# Recursive Sequences

There is a separate page with proofs. Sequences from the OEIS that pass the recursion test, but are not defined as recursions, are collected on a separate page: Non Recursions.

### Common properties:

• The number of words of length n in alphabet {1,2,...,d} avoiding words "11", "22", ..., "kk" is a recurrence sequence with initial terms a(0) = 1, a(1) = d and a recurrence relation a(n) = (d-1)a(n-1) + (d-k)a(n-2). Proof.
• The generating function f(x) = (mx+n)/(1 - dx - kx2) generates a sequence with the recurrence relation a(n) = da(n-1) + ka(n-2) and initial conditions a(0) = n and a(1) = nd + m.
• Let p and q be the roots of the equation x2 - dx - k = 0. Then the sequence a(n) = pn + qn = ((d+sqrt(d2+4k))/2)n + ((d-sqrt(d2+4k))/2)n satisfies the recurrence a(n) = da(n-1) + ka(n-2) with the initial conditions a(0) = 2, a(1) = d. Proof.
• Let p and q be the roots of the equation x2 - dx - k = 0. Then the sequence a(n) = (pn - qn)/(p-q) satisfies the recurrence a(n) = da(n-1) + ka(n-2) with the initial conditions a(0) = 0, a(1) = 1. Proof.

### a(n) = d * a(n-1) - a(n-2).

Common properties for sequences with initial terms a(0) = 1, a(1) = d-1:

• a(n) equals the number of 01-avoiding words of length n on alphabet {0,1,2, ..., d-1} which do not end in 0. Proof.
• a(n) equals the number of domino tilings in Sd-1 × P2n (product of a star graph and a path graph). Proof.
• The generating function of this sequence is (1-x)/(1-dx+x2).
• a(n+1)a(n-1) - a(n)2 = d-2. Proof.
• a(n+2)a(n-1) - a(n)a(n+1) = d(d-2). Proof.
• a(n) = (d-1) * a(n-1) + (d-2) * (a(n-2) + a(n-3) + ... + a(1) + a(0)). Proof.

Common properties for sequences with initial terms a(0) = 1, a(1) = d:

• The difference sequence follows the same recursion and is similar (shifted) to the sequence starting with a(0) = 1, a(1) = d-1.
• a(n) equals the number of 01-avoiding words of length n on alphabet {0,1,2, ..., d-1}. Proof.
• a(n) equals the number of domino tilings in Sd-1 × P2n+1 (product of a star graph and a path graph) with d-3 non central vertices removed from the last star. Proof.
• The generating function of this sequence is 1/(1-dx+x2).
• a(n) = (pn - qn)/(p-q), where p and q are the roots of the equation: x2 - dx + 1 = 0.
• a(n+1)a(n-1) - a(n)2 = -1. Proof.
• a(n+2)a(n-1) - a(n)a(n+1) = -d. Proof.

Common properties for sequences with initial terms a(0) = 2, a(1) = d:

• The generating function of this sequence is (2-dx)/(1-dx+x2).
• a(n) = pn + qn, where p and q are the roots of the equation: x2 - dx + 1 = 0. That is a(n) = ((d+sqrt(d2-4))/2)n + ((d-sqrt(d2-4))/2)n. Proof.
• Asymptotically a(n) = Round(pn), where p is the largest root of the equation: x2 - dx + 1 = 0. That is a(n) approaches ((d+sqrt(d2-4))/2)n. Proof.

Sequences:

• a(n) = - 14a(n-1) - a(n-2).
• Sequence: 1, 1, -15, 209, -2911, 40545, -564719,
In OEIS: - A122572 a(1)=a(2)=1, a(n)=-14a(n-1)-a(n-2)
• a(n) = - 3a(n-1) - a(n-2).
• Sequence: 1, -1, 1, -1, 1, -1, 1, -1,
In OEIS: - A033999 (-1)^n.
• Sequence: 1, -2, 5, -13, 34, -89, 233, -610,
In OEIS: - A099496 (-1)^nFib(2n+1).
• Sequence: -3, 11, -30, 79, -207, 542, -1419, 3715,
In OEIS: - A098150 a(n) = 2(a(n-2) - a(n-1)) + a(n-3) where a(0)=-3, a(1)=11 & a(2)=-30.
• Sequence: -1, -1, 4, -11, 29, -76, 199, -521, 1364,
In OEIS: - A098149 a(0)=-1, a(1)=-1, a(n)=-3*a(n-1)-a(n-2) for n > 1.
• a(n) = - 2a(n-1) - a(n-2).
• Sequence: 1, -1, 1, -1, 1, -1, 1, -1,
In OEIS: - A033999 (-1)^n.
• Sequence: 0, -1, 2, -3, 4, -5, 6, -7, 8, -9,
In OEIS: - A038608 n*(-1)^n.
• a(n) = - a(n-1) - a(n-2). This sequence is always periodic with period 3.
• Sequence: 1, 1, -2, 1, 1, -2, 1, 1, -2,
In OEIS: - A061347 Period 3.
• a(n) = a(n-1) - a(n-2). This sequence is always periodic with period 6.
• Sequence: 1, 2, 1, -1, -2, -1, 1, 2, 1,
In OEIS: - A057079 Periodic sequence 1,2,1,-1,-2,-1...; expansion of (1+x)/(1-x+x^2). Also A087204 Periodic sequence: 2,1,-1,-2,-1,1,...
• Sequence: 1, -2, -3, -1, 2, 3, 1, -2, -3,
In OEIS: - A117373 Expansion of (1-3x)/(1-x+x^2).
• Sequence: 1, 3, 2, -1, -3, -2, 1, 3, 2, -1,
In OEIS: - A119910 Simple periodic sequence with period 1,2,3,-1,-2,-3. submitted definition change
• Sequence: 1, -3, -4, -1, 3, 4, 1, -3, -4,
In OEIS: - A117378 Expansion of (1-4x)/(1-x+x^2).
• a(n) = 2a(n-1) - a(n-2). For d = 2 see arithmetic progressions.
• a(n) = 3a(n-1) - a(n-2).
• Sequence: 1, 2, 5, 13, 34, 89, 233, 610, 1597,
In OEIS: - A001519 a(n) = F(2n-1) = bisection of Fibonacci sequence. Also A122367 Dimension of 3-variable non-commutative harmonics (twisted derivative). The dimension of the space of non-commutative polynomials in 3 variables which are killed by all symmetric differential operators (where for a monomial w, d_{xi} ( xi w ) = w and d_{xi} ( xj w ) = 0 for i/=j). Also A048575 Pisot sequences L(2,5), E(2,5).
• Sequence: 1, 3, 8, 21, 55, 144, 377, 987, 2584,
In OEIS: - A001906 F(2n) = bisection of Fibonacci sequence.
• Sequence: 1, 4, 11, 29, 76, 199, 521, 1364, 3571,
In OEIS: - A002878 Bisection of Lucas sequence.
• Sequence: 1, 5, 14, 37, 97, 254, 665, 1741,
In OEIS: - A054486 A second order recursive sequence.
• Sequence: 1, 6, 17, 45, 118, 309, 809, 2118,
In OEIS: - A054492 a(n)=3a(n-1)-a(n-2), a(0)=1,a(0)=6.
• Sequence: 1, 7, 20, 53, 139, 364, 953, 2495,
In OEIS: - A055267 a(n)=3a(n-1)-a(n-2); a(0)=1, a(1)=7.
• Sequence: 1, 8, 23, 61, 160, 419, 1097, 2872,
In OEIS: - A055273 a(n)=3a(n-1)-a(n-2); a(0)=1, a(1)=8.
• Sequence: 1, 9, 26, 69, 181, 474, 1241, 3249, 8506,
In OEIS: - A055849 a(n)=3a(n-1)-a(n-2); a(0)=1, a(1)=9.
• Sequence: 1, 10, 29, 77, 202, 529, 1385, 3626,
In OEIS: - A055850 a(n)=3a(n-1)-a(n-2); a(0)=1, a(1)=10.
• Sequence: 1, 11, 32, 85, 223, 584, 1529, 4003,
In OEIS: - A056123 a(n)=3a(n-1)-a(n-2); a(0)=1, a(1)=11.
• Sequence: 2, 3, 7, 18, 47, 123, 322, 843, 2207,
In OEIS: - A005248 Bisection of Lucas numbers: A000032(2n).
• Sequence: 2, 6, 16, 42, 110, 288, 754, 1974, 5168,
In OEIS: - A025169 a(n)=2F(2n+2), where F=A000045 (the Fibonacci sequence).
• Sequence: 5, 10, 25, 65, 170, 445, 1165, 3050,
In OEIS: - A106729 Sum of two consecutive squares of Lucas numbers (A001254).
• Sequence: 7, 19, 50, 131, 343, 898, 2351, 6155,
In OEIS: - A100545 G.f.: (7-2x)/(x^2-3x+1).
• Sequence: 11, 15, 34, 87, 227, 594, 1555, 4071,
In OEIS: - A097512 6*Lucas(2n) - Fib(2n+2).
• a(n) = 4a(n-1) - a(n-2).
• Sequence: 1, 2, 7, 26, 97, 362, 1351, 5042,
In OEIS: - A001075 a(0) = 1, a(1) = 2, a(n) = 4a(n-1) - a(n-2).
• Sequence: 1, 3, 11, 41, 153, 571, 2131, 7953,
In OEIS: - A001835 a(n) = 4a(n-1) - a(n-2); a(0)=a(1)=1. Also A079935 a(n) = 4a(n-1) - a(n-2).
• Sequence: 1, 4, 15, 56, 209, 780, 2911,
In OEIS: - A001353 a(n) = 4a(n-1)-a(n-2) with a(0) = 0, a(1) = 1. Also A010905 Pisot sequence E(4,15), a(n)=[ a(n-1)^2/a(n-2)+1/2 ].
• Sequence: 1, 5, 19, 71, 265, 989, 3691, 13775,
In OEIS: - A001834 a(0) = 1, a(1) = 5, a(n) = 4a(n-1) - a(n-2).
• Sequence: 1, 6, 23, 86, 321, 1198, 4471, 16686,
In OEIS: - A054491 A second order recursive sequence.
• Sequence: 1, 7, 27, 101, 377, 1407, 5251, 19597,
In OEIS: - A054485 A second order recursive sequence.
• Sequence: 1, 8, 31, 116, 433, 1616, 6031, 22508,
In OEIS: - A055845 a(n)=4a(n-1)-a(n-2); a(0)=1, a(1)=8.
• Sequence: 2, 4, 14, 52, 194, 724, 2702, 10084,
In OEIS: - A003500 a(n) = 4a(n-1) - a(n-2).
• Sequence: 2, 9, 34, 127, 474, 1769, 6602, 24639,
In OEIS: - A077234 Bisection (odd part) of Chebyshev sequence with diophantine property.
• Sequence: 3, 9, 33, 123, 459, 1713, 6393, 23859,
In OEIS: - A082841 a(n)=4a(n-1)-a(n-2).
• Sequence: 0, 3, 12, 45, 168, 627, 2340, 8733,
In OEIS: - A005320 a(n) = 4a(n-1) - a(n-2).
• Sequence: 4, 9, 32, 119, 444, 1657, 6184, 23079,
In OEIS: - A057819 a(0)=4, a(1)=9, a(n)=4a(n-1)-a(n-2).
• Sequence: 4, 11, 40, 149, 556, 2075, 7744, 28901,
In OEIS: - A077236 Bisection (even part) of Chebyshev sequence with diophantine property.
• Sequence: 0, -1, -4, -15, -56, -209, -780, -2911, -10864,
In OEIS: - A106707 First entry of the vector (M^n)v, where M is the 2x2 matrix [[0,-1],[1,4]] and v is the column vector [0,1].
• Sequence: 5, 16, 59, 220, 821, 3064, 11435,
In OEIS: - A077235 Bisection (odd part) of Chebyshev sequence with diophantine property.
• a(n) = 5a(n-1) - a(n-2).
• Sequence: 1, 2, 9, 43, 206, 987, 4729, 22658,
In OEIS: - A002310 a(n) = 5*a(n-1) - a(n-2).
• Sequence: 1, 3, 14, 67, 321, 1538, 7369, 35307,
In OEIS: - A002320 a(n) = 5*a(n-1) - a(n-2).
• Sequence: 1, 4, 19, 91, 436, 2089, 10009, 47956,
In OEIS: - A004253 a(n) = 5a(n-1) - a(n-2).
• Sequence: 1, 5, 24, 115, 551, 2640, 12649,
In OEIS: - A004254 a(n) = 5a(n - 1) - a(n - 2), a(0) = 0, a(1) = 1.
• Sequence: 1, 6, 29, 139, 666, 3191, 15289,
In OEIS: - A030221 Chebyshev even indexed U-polynomials evaluated at sqrt(7)/2.
• Sequence: 1, 7, 34, 163, 781, 3742, 17929, 85903,
In OEIS: - A055271 a(n)=5a(n-1)-a(n-2); a(0)=1, a(1)=7.
• Sequence: 1, 9, 44, 211, 1011, 4844, 23209, 111201,
In OEIS: - A099867 a(n) = 5a(n - 1) - a(n - 2), a(0) = 1, a(1) = 9.
• Sequence: 1, 13, 64, 307, 1471, 7048, 33769, 161797,
In OEIS: - A054477 A Pellian-related sequence.
• Sequence: 2, 5, 23, 110, 527, 2525, 12098, 57965,
In OEIS: - A003501 a(n) = 5a(n-1) - a(n-2).
• Sequence: 3, 25, 122, 585, 2803, 13430, 64347,
In OEIS: - A099868 a(n) = 5a(n - 1) - a(n - 2), a(0) = 3, a(1) = 25.
• a(n) = 6a(n-1) - a(n-2).
• Sequence: 1, 2, 11, 64, 373, 2174, 12671, 73852, 430441,
In OEIS: - A038725 A second order recursive sequence.
• Sequence: 1, 3, 17, 99, 577, 3363, 19601, 114243,
In OEIS: - A001541 a(0) = 1, a(1) = 3; for n > 1, a(n) = 6a(n-1) - a(n-2).
• Sequence: 1, 4, 23, 134, 781, 4552, 26531, 154634,
In OEIS: - A038723 A second order recursive sequence.
• Sequence: 1, 5, 29, 169, 985, 5741, 33461, 195025,
In OEIS: - A001653 Numbers n such that 2*n^2 - 1 is a square.
• Sequence: 1, 6, 35, 204, 1189, 6930, 40391,
In OEIS: - A001109 a(n)^2 is a triangular number: a(n) = 6*a(n-1) - a(n-2) with a(0)=0, a(1)=1.
• Sequence: 1, 7, 41, 239, 1393, 8119, 47321, 275807,
In OEIS: - A002315 NSW numbers: a(n) = 6*a(n-1) - a(n-2); also a(n)^2 - 2*b(n)^2 = -1 with b(n)=A001653(n).
• Sequence: 1, 8, 47, 274, 1597, 9308, 54251, 316198,
In OEIS: - A054488 A second order recursive sequence.
• Sequence: 1, 9, 53, 309, 1801, 10497, 61181, 356589,
In OEIS: - A038761 a(n)=6a(n-1)-a(n-2), n >= 2, a(0)=1, a(1)=9.
• Sequence: 1, 10, 59, 344, 2005, 11686, 68111, 396980,
In OEIS: - A054489 A second order recursive sequence.
• Sequence: 1, 11, 65, 379, 2209, 12875, 75041,
In OEIS: - A054490 A Pellian-related second order recursive sequence.
• Sequence: 2, 12, 70, 408, 2378, 13860, 80782,
In OEIS: - A001542 a(n) = 6a(n-1) - a(n-2).
• Sequence: 2, 6, 34, 198, 1154, 6726, 39202, 228486,
In OEIS: - A003499 a(0) = 2, a(1) = 6; for n >= 2, a(n) = 6a(n-1) - a(n-2).
• Sequence: 2, 10, 58, 338, 1970, 11482, 66922, 390050,
In OEIS: - A075870 2*n^2 - 4 is a square.
• Sequence: 2, 13, 76, 443, 2582, 15049, 87712, 511223,
In OEIS: - A077413 Bisection (odd part) of Chebyshev sequence with diophantine property.
• Sequence: 2, 14, 82, 478, 2786, 16238, 94642, 551614,
In OEIS: - A077444 Numbers n such that (n^2 + 4)/2 is a square.
• Sequence: 3, 9, 51, 297, 1731, 10089, 58803, 342729,
In OEIS: - A106329 Numbers k such that k^2 = 8*(j^2) + 9.
• Sequence: 3, 13, 75, 437, 2547, 14845, 86523,
In OEIS: - A038762 A Pellian-related sequence.
• Sequence: 3, 15, 87, 507, 2955, 17223, 100383,
In OEIS: - A075841 2*n^2 - 9 is a square.
• Sequence: 3, 18, 105, 612, 3567, 20790, 121173,
In OEIS: - A106328 Numbers j such that 8*(j^2) + 9 = k^2 for some positive number k.
• Sequence: 4, 20, 116, 676, 3940, 22964, 133844, 780100,
In OEIS: - A077445 Numbers n such that (n^2 - 8)/2 is a square.
• Sequence: 4, 22, 128, 746, 4348, 25342, 147704,
In OEIS: - A100525 Bisection of A048654.
• Sequence: 0, 4, 24, 140, 816, 4756, 27720, 161564,
In OEIS: - A005319 a(n) = 6a(n-1) - a(n-2).
• Sequence: 5, 23, 133, 775, 4517, 26327, 153445,
In OEIS: - A077240 Bisection (even part) of Chebyshev sequence with diophantine property.
• Sequence: 5, 27, 157, 915, 5333, 31083, 181165,
In OEIS: - A101386 G.f.: (5-3x)/(x^2-6x+1).
• Sequence: 0, 6, 36, 210, 1224, 7134, 41580, 242346,
In OEIS: - A075848 2*n^2 + 9 is a square.
• Sequence: 7, 37, 215, 1253, 7303, 42565, 248087,
In OEIS: - A077239 Bisection (odd part) of Chebyshev sequence with diophantine property.
• Sequence: 0, 8, 48, 280, 1632, 9512, 55440, 323128,
In OEIS: - A081554 a(n)=sqrt(2)((3+2sqrt(2))^n-(3-2sqrt(2))^n).
• a(n) = 7a(n-1) - a(n-2).
• Sequence: 1, 5, 34, 233, 1597, 10946, 75025, 514229,
In OEIS: - A033889 Fibonacci(4n+1).
• Sequence: 1, 6, 41, 281, 1926, 13201, 90481, 620166,
In OEIS: - A049685 a(n)=L(4n+2)/3, where L=A000032 (the Lucas sequence).
• Sequence: 1, 7, 48, 329, 2255, 15456, 105937,
In OEIS: - A004187 a(n) = 7*a(n-1) - a(n-2).
• Sequence: 1, 8, 55, 377, 2584, 17711, 121393,
In OEIS: - A033890 Fibonacci(4n+2).
• Sequence: 1, 11, 76, 521, 3571, 24476, 167761,
In OEIS: - A056914 a(n)=L(4n+1) where L() are the Lucas numbers.
• Sequence: 2, 7, 47, 322, 2207, 15127, 103682,
In OEIS: - A056854 a(n)=7a(n-1)-a(n-2), a(0)=2, a(1)=7.
• Sequence: 2, 13, 89, 610, 4181, 28657, 196418,
In OEIS: - A033891 Fibonacci(4n+3).
• Sequence: 3, 21, 144, 987, 6765, 46368, 317811,
In OEIS: - A033888 Fibonacci(4n).
• a(n) = 8a(n-1) - a(n-2).
• Sequence: 1, 4, 31, 244, 1921, 15124, 119071,
In OEIS: - A001091 a(n) = 8a(n-1) - a(n-2); a(0) = 1, a(1) = 4.
• Sequence: 1, 5, 39, 307, 2417, 19029, 149815,
In OEIS: - A105426 a(0)=1, a(1)=5, a(n)=8*a(n-1)-a(n-2).
• Sequence: 1, 7, 55, 433, 3409, 26839, 211303, 1663585,
In OEIS: - A070997 a(n) = 8*a(n-1) - a(n-2), a(0)=1, a(-1)=1.
• Sequence: 1, 8, 63, 496, 3905, 30744, 242047,
In OEIS: - A001090 a(n) = 8*a(n-1)-a(n-2); a(0) = 0, a(1) = 1.
• Sequence: 1, 9, 71, 559, 4401, 34649, 272791,
In OEIS: - A057080 Even indexed Chebyshev U-polynomials evaluated at sqrt(10)/2.
• Sequence: 1, 10, 79, 622, 4897, 38554, 303535,
In OEIS: - A077245 Bisection (even part) of Chebyshev sequence with diophantine property.
• Sequence: 2, 8, 62, 488, 3842, 30248, 238142,
In OEIS: - A086903 a(n) = 8a(n-1) - a(n-2), starting with a(0) = 2 and a(1) = 8, a(n) = (4+sqrt(15))^n + (4-sqrt(15))^n.
• Sequence: 2, 13, 102, 803, 6322, 49773, 391862,
In OEIS: - A077246 Bisection (even part) of Chebyshev sequence with diophantine property.
• Sequence: 2, 17, 134, 1055, 8306, 65393, 514838,
In OEIS: - A077243 Bisection (odd part) of Chebyshev sequence with diophantine property.
• Sequence: 3, 22, 173, 1362, 10723, 84422, 664653,
In OEIS: - A077244 Bisection (odd part) of Chebyshev sequence with diophantine property.
• a(n) = 9a(n-1) - a(n-2).
• Sequence: 1, 8, 71, 631, 5608, 49841, 442961, 3936808,
In OEIS: - A070998 a(n) = 9*a(n-1) - a(n-2), a(0)=1, a(-1)=1.
• Sequence: 1, 9, 80, 711, 6319, 56160, 499121,
In OEIS: - A018913 a(n) = 9a(n - 1) - a(n - 2); a(0) = 0, a(1) = 1.
• Sequence: 1, 10, 89, 791, 7030, 62479, 555281,
In OEIS: - A057081 Even indexed Chebyshev U-polynomials evaluated at sqrt(11)/2.
• Sequence: 2, 9, 79, 702, 6239, 55449, 492802,
In OEIS: - A056918 a(n)=9*a(n-1)-a(n-2); a(0)=2, a(1)=9.
• Sequence: 3, 27, 240, 2133, 18957, 168480, 1497363,
In OEIS: - A065100 a(0) = c, a(1) = p*c^3; a(n+2) = p*c^2*a(n+1) - a(n), for p = 1, c = 3.
• a(n) = 10a(n-1) - a(n-2).
• Sequence: 0, 2, 20, 198, 1960, 19402, 192060, 1901198,
In OEIS: - A001078 a(n) = 10*a(n-1)-a(n-2) with a(0) = 0, a(1) = 2.
• Sequence: 1, 5, 49, 485, 4801, 47525, 470449,
In OEIS: - A001079 a(n) = 10a(n-1) - a(n-2); a(0) = 1, a(1) = 5.
• Sequence: 1, 9, 89, 881, 8721, 86329, 854569,
In OEIS: - A072256 a(n) = 10*a(n-1) - a(n-2); a(0) = a(1) = 1.
• Sequence: 1, 10, 99, 980, 9701, 96030, 950599,
In OEIS: - A004189 a(n) = 10*a(n-1)-a(n-2); a(0) = 0, a(1) = 1.
• Sequence: 1, 11, 109, 1079, 10681, 105731, 1046629,
In OEIS: - A054320 G.f.: (1+x)/(1-10*x+x^2).
• Sequence: 1, 12, 119, 1178, 11661, 115432, 1142659,
In OEIS: - A077251 Bisection (even part) of Chebyshev sequence with diophantine property.
• Sequence: 2, 10, 98, 970, 9602, 95050, 940898,
In OEIS: - A087799 a(n) =10a(n-1) - a(n-2), starting with a(0) = 2 and a(1) = 10, a(n) = (5+sqrt(24))^n + (5-sqrt(24))^n.
• Sequence: 2, 21, 208, 2059, 20382, 201761, 1997228,
In OEIS: - A077249 Bisection (odd part) of Chebyshev sequence with diophantine property.
• Sequence: 0, 4, 40, 396, 3920, 38804, 384120, 3802396,
In OEIS: - A122652 a(0)=0, a(1)=4, a(n)=10*a(n-1)-a(n-2).
• Sequence: 0, 6, 60, 594, 5880, 58206, 576180,
In OEIS: - A122653 a(0)=0, a(1)=6, a(n)=10*a(n-1)-a(n-2).
• Sequence: 7, 59, 583, 5771, 57127, 565499, 5597863,
In OEIS: - A077409 Bisection (even part) of Chebyshev sequence with diophantine property.
• Sequence: 11, 103, 1019, 10087, 99851, 988423, 9784379,
In OEIS: - A077250 Bisection (odd part) of Chebyshev sequence with diophantine property.
• a(n) = 11a(n-1) - a(n-2).
• Sequence: 1, 10, 109, 1189, 12970, 141481, 1543321,
In OEIS: - A078922 a(n) = 11*a(n-1) - a(n-2).
• Sequence: 1, 11, 120, 1309, 14279, 155760, 1699081,
In OEIS: - A004190 Expansion of 1/(1-11*x+x^2).
• Sequence: 1, 12, 131, 1429, 15588, 170039, 1854841,
In OEIS: - A097783 Chebyshev polynomials S(n,11) + S(n-1,11) with diophantine property.
• Sequence: 2, 11, 119, 1298, 14159, 154451, 1684802,
In OEIS: - A057076 A Chebyshev or generalized Fibonacci sequence.
• Sequence: 3, 33, 360, 3927, 42837, 467280, 5097243,
In OEIS: - A075835 13*n^2 + 4 is a square.
• a(n) = 12a(n-1) - a(n-2).
• Sequence: 1, 6, 71, 846, 10081, 120126, 1431431,
In OEIS: - A023038 a(n) = 12a(n-1) - a(n-2).
• Sequence: 1, 11, 131, 1561, 18601, 221651, 2641211,
In OEIS: - A077417 Chebyshev T-sequence with diophantine property.
• Sequence: 1, 12, 143, 1704, 20305, 241956, 2883167,
In OEIS: - A004191 Expansion of 1/(1-12*x+x^2).
• Sequence: 1, 13, 155, 1847, 22009, 262261, 3125123,
In OEIS: - A077416 Chebyshev S-sequence with diophantine property.
• Sequence: 2, 12, 142, 1692, 20162, 240252, 2862862,
In OEIS: - A087800 a(n) =12a(n-1) - a(n-2), starting with a(0) = 2 and a(1) = 12, a(n) = (6+sqrt(35))^n + (6-sqrt(35))^n.
• Sequence: 2, 24, 286, 3408, 40610, 483912, 5766334,
In OEIS: - A065101 a(0) = c, a(1) = p*c^3; a(n+2) = p*c^2*a(n+1) - a(n), for p = 3, c = 2.
• a(n) = 13a(n-1) - a(n-2).
• Sequence: 1, 12, 155, 2003, 25884, 334489, 4322473,
In OEIS: - A085260 Ratio-determined insertion sequence I(0.0833344) (see the link below).
• Sequence: 1, 13, 168, 2171, 28055, 362544, 4685017,
In OEIS: - A078362 A Chebyshev S-sequence with diophantine property.
• Sequence: 2, 13, 167, 2158, 27887, 360373, 4656962,
In OEIS: - A078363 A Chebyshev T-sequence with diophantine property.
• a(n) = 14a(n-1) - a(n-2).
• Sequence: 1, 7, 97, 1351, 18817, 262087, 3650401,
In OEIS: - A011943 Numbers n such that any group of n consecutive integers has integral standard deviation {viz. A011944(n)}.
• Sequence: 1, 11, 153, 2131, 29681, 413403,
In OEIS: - A122769 Numbers n such that n^2 is of the form 1+2m+3m^2 (A056109).
• Sequence: 1, 13, 181, 2521, 35113, 489061,
In OEIS: - A001570 Numbers n such that n^2 is simultaneously square and centered hexagonal. Also A122571 a(1)=a(2)=1, a(n)=14a(n-1)-a(n-2).
• Sequence: 1, 14, 195, 2716, 37829, 526890,
In OEIS: - A007655 Standard deviation of A007654.
• Sequence: 1, 15, 209, 2911, 40545, 564719, 7865521,
In OEIS: - A028230 Bisection of A001353. Indices of square numbers which are also octagonal.
• Sequence: 2, 2, 26, 362, 5042, 70226, 978122,
In OEIS: - A094347 a(n) = 14*a(n-1)-a(n-2); a(0) = a(1) = 2.
• Sequence: 2, 12, 142, 1692, 20162, 240252, 2862862,
In OEIS: - A087800 a(n) =12a(n-1) - a(n-2), starting with a(0) = 2 and a(1) = 12, a(n) = (6+sqrt(35))^n + (6-sqrt(35))^n.
• Sequence: 2, 14, 194, 2702, 37634, 524174, 7300802,
In OEIS: - A067902 a(n) = 14*a(n-1) - a(n-2); a(0) = 2, a(1) = 14.
• Sequence: 0, 2, 28, 390, 5432, 75658, 1053780,
In OEIS: - A011944 a(n) = 14*a(n-1)-a(n-2) with a(0) = 0, a(1) = 2.
• Sequence: 0, 6, 84, 1170, 16296, 226974, 3161340,
In OEIS: - A011945 Area of triangles with integral side lengths a-1, a, a+1 and integral area.
• Sequence: 0, 8, 112, 1560, 21728, 302632, 4215120,
In OEIS: - A067900 a(n) = 14*a(n-1) - a(n-2); a(0) = 0, a(1) = 8.
• a(n) = 15a(n-1) - a(n-2).
• Sequence: 1, 15, 224, 3345, 49951, 745920, 11138849,
In OEIS: - A078364 A Chebyshev S-sequence with diophantine property.
• Sequence: 2, 15, 223, 3330, 49727, 742575, 11088898,
In OEIS: - A078365 A Chebyshev T-sequence with diophantine property.
• a(n) = 16a(n-1) - a(n-2).
• Sequence: 1, 8, 127, 2024, 32257, 514088, 8193151,
In OEIS: - A001081 a(n) = 16a(n-1) - a(n-2).
• Sequence: 1, 16, 255, 4064, 64769, 1032240, 16451071,
In OEIS: - A077412 Chebyshev U(n,x) polynomial evaluated at x=8.
• Sequence: 0, 3, 48, 765, 12192, 194307, 3096720,
In OEIS: - A001080 a(n) = 16*a(n-1)-a(n-2) with a(0) = 0, a(1) = 3.
• Sequence: 2, 16, 254, 4048, 64514, 1028176, 16386302,
In OEIS: - A090727 a(n) = 16a(n-1) - a(n-2), starting with a(0) = 2 and a(1) = 16.
• a(n) = 17a(n-1) - a(n-2).
• Sequence: 1, 17, 288, 4879, 82655, 1400256, 23721697,
In OEIS: - A078366 A Chebyshev S-sequence with diophantine property.
• Sequence: 2, 17, 287, 4862, 82367, 1395377, 23639042,
In OEIS: - A078367 A Chebyshev T-sequence with diophantine property.
• a(n) = 18a(n-1) - a(n-2).
• Sequence: 1, 9, 161, 2889, 51841, 930249, 16692641,
In OEIS: - A023039 a(n) = 18a(n-1) - a(n-2).
• Sequence: 1, 17, 305, 5473, 98209, 1762289, 31622993,
In OEIS: - A007805 a(n)=F(6n+3)/2, where F=A000045 (the Fibonacci sequence).
• Sequence: 1, 18, 323, 5796, 104005, 1866294, 33489287,
In OEIS: - A049660 a(n)=F(6n)/8, where F=A000045 (the Fibonacci sequence).
• Sequence: 1, 19, 341, 6119, 109801, 1970299, 35355581,
In OEIS: - A049629 a(n)=(F(6n+5)-F(6n+1))/4=(F(6n+4)+F(6n+2))/4, where F=A000045 (the Fibonacci sequence).
• Sequence: 2, 18, 322, 5778, 103682, 1860498, 33385282,
In OEIS: - A087215 Lucas(6n): a(n) = 18a(n-1) - a(n-2), starting with a(0) = 2 and a(1) = 18.
• Sequence: 2, 38, 682, 12238, 219602, 3940598, 70711162,
In OEIS: - A075796 5*x^2 + 5 is a square.
• Sequence: 3, 51, 915, 16419, 294627, 5286867,
In OEIS: - A075869 5*n^2 - 9 is a square.
• Sequence: 3, 54, 969, 17388, 312015, 5598882, 100467861,
In OEIS: - A065102 a(0) = c, a(1) = p*c^3; a(n+2) = p*c^2*a(n+1) - a(n), for p = 2, c = 3.
• Sequence: 3, 55, 987, 17711, 317811, 5702887, 102334155,
In OEIS: - A103134 Fib(6n+4).
• Sequence: 0, 4, 72, 1292, 23184, 416020, 7465176,
In OEIS: - A060645 a(0) = 0, a(1) = 4 then a(n) = 18*a(n-1)-a(n-2).
• a(n) = 19a(n-1) - a(n-2).
• Sequence: 1, 19, 360, 6821, 129239, 2448720, 46396441,
In OEIS: - A078368 A Chebyshev S-sequence with diophantine property..
• Sequence: 2, 19, 359, 6802, 128879, 2441899, 46267202,
In OEIS: - A078369 A Chebyshev T-sequence with diophantine property.
• a(n) = 20a(n-1) - a(n-2).
• Sequence: 0, 3, 60, 1197, 23880, 476403, 9504180,
In OEIS: - A001084 a(n) = 20*a(n-1)-a(n-2) with a(0) = 0, a(1) = 3.
• Sequence: 1, 10, 199, 3970, 79201, 1580050, 31521799,
In OEIS: - A001085 a(n) = 20a(n-1) - a(n-2).
• Sequence: 1, 19, 379, 7561, 150841, 3009259, 60034339,
In OEIS: - A075839 11*n^2 - 2 is a square.
• Sequence: 1, 20, 399, 7960, 158801, 3168060,
In OEIS: - A075843 99*a(n)^2 + 1 is a square.
• Sequence: 1, 21, 419, 8359, 166761, 3326861,
In OEIS: - A083043 Integers y such that 11x^2-9y^2=2 for some integer x.
• Sequence: 2, 20, 398, 7940, 158402, 3160100,
In OEIS: - A090728 a(n) = 20a(n-1) - a(n-2), starting with a(0) = 2 and a(1) = 20.
• Sequence: 0, 6, 120, 2394, 47760, 952806, 19008360,
In OEIS: - A075844 11*n^2 + 4 is a square.
• a(n) = 21a(n-1) - a(n-2).
• Sequence: 1, 21, 440, 9219, 193159, 4047120,
In OEIS: - A092499 Chebyshev polynomials S(n-1,21) with diophantine property.
• Sequence: 2, 21, 439, 9198, 192719, 4037901,
In OEIS: - A090729 a(n) = 21a(n-1) - a(n-2), starting with a(0) = 2 and a(1) = 21.
• a(n) = 22a(n-1) - a(n-2).
• Sequence: 1, 11, 241, 5291, 116161, 2550251,
In OEIS: - A077422 Chebyshev sequence T(n,11) with diophantine property.
• Sequence: 1, 22, 483, 10604, 232805, 5111106,
In OEIS: - A077421 Chebyshev sequence U(n,11)=S(n,22) with diophantine property.
• Sequence: 2, 22, 482, 10582, 232322, 5100502,
In OEIS: - A090730 a(n) = 22a(n-1) - a(n-2), starting with a(0) = 2 and a(1) = 22.
• a(n) = 23a(n-1) - a(n-2).
• Sequence: 1, 23, 528, 12121, 278255, 6387744,
In OEIS: - A097778 Chebyshev polynomials S(n,23) with diophantine property.
• Sequence: 2, 23, 527, 12098, 277727, 6375623,
In OEIS: - A090731 a(n) = 23a(n-1) - a(n-2), starting with a(0) = 2 and a(1) = 23.
• a(n) = 24a(n-1) - a(n-2).
• Sequence: 1, 12, 287, 6876, 164737, 3946812, 94558751,
In OEIS: - A077424 Chebyshev sequence T(n,12) with diophantine property.
• Sequence: 1, 24, 575, 13776, 330049, 7907400,
In OEIS: - A077423 Chebyshev sequence U(n,12)=S(n,24) with diophantine property.
• Sequence: 2, 24, 574, 13752, 329474, 7893624,
In OEIS: - A090732 a(n) = 24a(n-1) - a(n-2), starting with a(0) = 2 and a(1) = 24.
• a(n) = 25a(n-1) - a(n-2).
• Sequence: 1, 25, 624, 15575, 388751, 9703200,
In OEIS: - A097780 Chebyshev polynomials S(n,25) with diophantine property.
• Sequence: 2, 25, 623, 15550, 388127, 9687625,
In OEIS: - A090733 a(n) = 25a(n-1) - a(n-2), starting with a(0) = 2 and a(1) = 25.
• a(n) = 26a(n-1) - a(n-2).
• Sequence: 1, 13, 337, 8749, 227137, 5896813,
In OEIS: - A097308 Chebyshev T-polynomials T(n,13) with diophantine property.
• Sequence: 1, 26, 675, 17524, 454949, 11811150,
In OEIS: - A097309 Chebyshev polynomials of the second kind, U(n,x), evaluated at x=13.
• Sequence: 2, 26, 674, 17498, 454274, 11793626,
In OEIS: - A090247 a(n) = 26a(n-1) - a(n-2), starting with a(0) = 2 and a(1) = 26.
• a(n) = 27a(n-1) - a(n-2).
• Sequence: 1, 26, 701, 18901, 509626, 13741001,
In OEIS: - A097835 First differences of Chebyshev polynomials S(n,27)=A097781(n) with diophantine property.
• Sequence: 1, 27, 728, 19629, 529255, 14270256,
In OEIS: - A097781 Chebyshev polynomials S(n,27) with diophantine property.
• Sequence: 1, 28, 755, 20357, 548884, 14799511,
In OEIS: - A097834 Chebyshev polynomials S(n,27) + S(n-1,27) with diophantine property.
• Sequence: 2, 27, 727, 19602, 528527, 14250627,
In OEIS: - A090248 a(n) =27a(n-1) - a(n-2), starting with a(0) = 2 and a(1) = 27.
• a(n) = 28a(n-1) - a(n-2).
• Sequence: 1, 14, 391, 10934, 305761, 8550374,
In OEIS: - A097310 Chebyshev T-polynomials T(n,14) with diophantine property.
• Sequence: 1, 28, 783, 21896, 612305, 17122644,
In OEIS: - A097311 Chebyshev polynomials of the second kind, U(n,x), evaluated at x=14.
• Sequence: 2, 28, 782, 21868, 611522, 17100748,
In OEIS: - A090249 a(n) =28a(n-1) - a(n-2), starting with a(0) = 2 and a(1) = 28.
• a(n) = 29a(n-1) - a(n-2).
• Sequence: 1, 29, 840, 24331, 704759, 20413680,
In OEIS: - A097782 Chebyshev polynomials S(n,29) with diophantine property.
• Sequence: 2, 29, 839, 24302, 703919, 20389349,
In OEIS: - A090251 a(n) =29a(n-1) - a(n-2), starting with a(0) = 2 and a(1) = 29.
• a(n) = 30a(n-1) - a(n-2).
• Sequence: 1, 15, 449, 13455, 403201, 12082575,
In OEIS: - A068203 Chebyshev T-polynomials T(n,15) with diophantine property.
• Sequence: 1, 30, 899, 26940, 807301, 24192090,
In OEIS: - A097313 Chebyshev polynomials of the second kind, U(n,x), evaluated for x=15.
• Sequence: 4, 120, 3596, 107760, 3229204, 96768360,
In OEIS: - A068204 Let (x_n, y_n) be n-th solution to the Pell equation x^2 = 14*y^2 + 1. Sequence gives {y_n}.
• a(n) = 31a(n-1) - a(n-2).
• Sequence: 1, 30, 929, 28769, 890910, 27589441,
In OEIS: - A111216 a(n)=31*a(n-1)-a(n-2).
• a(n) = 32a(n-1) - a(n-2).
• Sequence: 1, 32, 1023, 32704, 1045505, 33423456,
In OEIS: - A029548 Expansion of 1/(1-32*x+x^2).
• a(n) = 34a(n-1) - a(n-2).
• Sequence: 1, 17, 577, 19601, 665857, 22619537,
In OEIS: - A056771 a(n)=a(-n)=34a(n-1)-a(n-2) and a(0)=1.
• Sequence: 1, 33, 1121, 38081, 1293633, 43945441,
In OEIS: - A077420 Bisection of Chebyshev sequence T(n,3) (odd part) with diophantine property.
• Sequence: 1, 34, 1155, 39236, 1332869, 45278310,
In OEIS: - A029547 Expansion of 1/(1-34*x+x^2). Also A091761 Pell(4n)/Pell(4).
• Sequence: 1, 35, 1189, 40391, 1372105, 46611179,
In OEIS: - A046176 Indices of square numbers which are also hexagonal.
• Sequence: 1, 38, 1291, 43856, 1489813, 50609786,
In OEIS: - A027657 Expansion of (1+4*x)/(1-34*x+x^2).
• Sequence: 0, 6, 204, 6930, 235416, 7997214, 271669860,
In OEIS: - A082405 a(n) = 34*a(n-1) - a(n-2); a(0)=0, a(1)=6.
• a(n) = 38a(n-1) - a(n-2).
• Sequence: 1, 19, 721, 27379, 1039681, 39480499,
In OEIS: - A078986 Chebyshev T(n,19) polynomial.
• Sequence: 1, 37, 1405, 53353, 2026009, 76934989,
In OEIS: - A097315 Pell equation solutions (3*b(n))^2 - 10*a(n)^2 = -1 with b(n):=A097314(n), n ≥ 0.
• Sequence: 1, 38, 1443, 54796, 2080805, 79015794,
In OEIS: - A078987 Chebyshev U(n,x) polynomial evaluated at x=19.
• Sequence: 1, 39, 1481, 56239, 2135601, 81096599,
In OEIS: - A097314 Pell equation solutions (3*a(n))^2 - 10*b(n)^2 = -1 with b(n):=A097315(n), n>=0.
• Sequence: 0, 6, 228, 8658, 328776, 12484830, 474094764,
In OEIS: - A084070 a(0)=0, a(1)=6, a(n)=38*a(n-1)-a(n-2).
• a(n) = 47a(n-1) - a(n-2).
• Sequence: 1, 41, 1926, 90481, 4250681, 199691526,
In OEIS: - A049676 a(n)=(F(8n+3)+F(8n+1))/3, where F=A000045 (the Fibonacci sequence).
• Sequence: 1, 47, 2208, 103729, 4873055, 228929856,
In OEIS: - A049668 a(n)=F(8n)/21, where F=A000045 (the Fibonacci sequence).
• Sequence: 1, 48, 2255, 105937, 4976784, 233802911,
In OEIS: - A049678 a(n)=F(8n+4)/3, where F=A000045 (the Fibonacci sequence)..
• Sequence: 2, 47, 2207, 103682, 4870847, 228826127,
In OEIS: - A087265 Lucas numbers L(8n).
• Sequence: 3, 137, 6436, 302355, 14204249, 667297348,
In OEIS: - A049677 a(n)=(F(8n+6)+F(8n+1))/3, where F=A000045 (the Fibonacci sequence).
• Sequence: 6, 281, 13201, 620166, 29134601, 1368706081,
In OEIS: - A049679 a(n)=(F(8n+7)+F(8n+5))/3, where F=A000045 (the Fibonacci sequence).
• Sequence: 0, 7, 329, 15456, 726103, 34111385,
In OEIS: - A049686 a(n)=F(8n)/3, where F=A000045 (the Fibonacci sequence).
• a(n) = 48a(n-1) - a(n-2).
• Sequence: 1, 24, 1151, 55224, 2649601, 127125624,
In OEIS: - A114051 x such that x^2 - 23*y^2 = 1.
• a(n) = 51a(n-1) - a(n-2).
• Sequence: 1, 50, 2549, 129949, 6624850, 337737401,
In OEIS: - A097838 First differences of Chebyshev polynomials S(n,51)=A097836(n) with diophantine property.
• Sequence: 1, 51, 2600, 132549, 6757399, 344494800,
In OEIS: - A097836 Chebyshev polynomials S(n,51).
• Sequence: 1, 52, 2651, 135149, 6889948, 351252199,
In OEIS: - A097837 Chebyshev polynomials S(n,51) + S(n-1,51) with diophantine property.
• Sequence: 2, 51, 2599, 132498, 6754799, 344362251,
In OEIS: - A099368 Twice Chebyshev's polynomials of the first kind, T(n,x), evaluated at x=51/2.
• a(n) = 52a(n-1) - a(n-2).
• Sequence: 1, 26, 1351, 70226, 3650401, 189750626,
In OEIS: - A114052 x such that x^2 - 27*y^2 = 1.
• a(n) = 66a(n-1) - a(n-2).
• Sequence: 1, 33, 2177, 143649, 9478657, 625447713,
In OEIS: - A099370 Chebyshev's polynomial of the first kind, T(n,x), evaluated at x=33.
• Sequence: 1, 65, 4289, 283009, 18674305, 1232221121,
In OEIS: - A078988 Chebyshev sequence with diophantine property.
• Sequence: 1, 66, 4355, 287364, 18961669, 1251182790,
In OEIS: - A097316 Chebyshev U(n,x) polynomial evaluated at x=33.
• Sequence: 1, 67, 4421, 291719, 19249033, 1270144459,
In OEIS: - A078989 Chebyshev sequence with diophantine property.
• Sequence: 0, 8, 528, 34840, 2298912, 151693352,
In OEIS: - A121740 Solutions to the Pell equation x^2 - 17y^2 = 1 (y values).
• a(n) = 83a(n-1) - a(n-2).
• Sequence: 1, 82, 6805, 564733, 46866034, 3889316089,
In OEIS: - A097841 First differences of Chebyshev polynomials S(n,83)=A097839(n) with diophantine property.
• Sequence: 1, 83, 6888, 571621, 47437655, 3936753744,
In OEIS: - A097839 Chebyshev polynomials S(n,83).
• Sequence: 1, 84, 6971, 578509, 48009276, 3984191399,
In OEIS: - A097840 Chebyshev polynomials S(n,83) + S(n-1,83) with diophantine property.
• Sequence: 2, 83, 6887, 571538, 47430767, 3936182123,
In OEIS: - A099373 Twice Chebyshev's polynomials of the first kind, T(n,x), evaluated at 83/2.
• a(n) = 98a(n-1) - a(n-2).
• Sequence: 1, 99, 9701, 950599, 93149001, 9127651499,
In OEIS: - A046173 Indices of square numbers which are also pentagonal.
• Sequence: 0, 20, 1960, 192060, 18819920, 1844160100,
In OEIS: - A072818 Possibly the only integers of the form sqrt(m^2*(m^2-1)*2/3) [only checked for the first 5 terms].
• a(n) = 102a(n-1) - a(n-2).
• Sequence: 1, 51, 5201, 530451, 54100801, 5517751251,
In OEIS: - A099397 Chebyshev's polynomial of the first kind, T(n,x), evaluated at x=51.
• Sequence: 1, 101, 10301, 1050601, 107151001, 10928351501,
In OEIS: - A097727 Pell equation solutions (5*b(n))^2 - 26*a(n)^2 = -1 with b(n):=A097726(n), n ≥ 0.
• Sequence: 1, 102, 10403, 1061004, 108212005, 11036563506,
In OEIS: - A097725 Chebyshev U(n,x) polynomial evaluated at x=51.
• Sequence: 1, 103, 10505, 1071407, 109273009, 11144775511,
In OEIS: - A097726 Pell equation solutions (5*a(n))^2 - 26*b(n)^2 = -1 with b(n):=A097727(n), n ≥ 0.
• a(n) = 110a(n-1) - a(n-2).
• Sequence: 1, 55, 6049, 665335, 73180801, 8049222775,
In OEIS: - A114049 x such that x^2 - 21*y^2 = 1.
• a(n) = 123a(n-1) - a(n-2).
• Sequence: 1, 122, 15005, 1845493, 226980634, 27916772489,
In OEIS: - A097843 First differences of Chebyshev polynomials S(n,123)=A049670(n+1) with diophantine property.
• Sequence: 1, 123, 15128, 1860621, 228841255, 28145613744,
In OEIS: - A049670 a(n)=F(10n)/55, where F=A000045 (the Fibonacci sequence).
• Sequence: 1, 124, 15251, 1875749, 230701876, 28374454999,
In OEIS: - A097842 Chebyshev polynomials S(n,123) + S(n-1,123) with diophantine property.
• Sequence: 2, 123, 15127, 1860498, 228826127, 28143753123,
In OEIS: - A065705 Lucas numbers L(10n).
• a(n) = 146a(n-1) - a(n-2).
• Sequence: 1, 145, 21169, 3090529, 451196065, 65871534961,
In OEIS: - A097730 Pell equation solutions (6*b(n))^2 - 37*a(n)^2 = -1 with b(n):=A097729(n), n ≥ 0.
• Sequence: 1, 146, 21315, 3111844, 454307909, 66325842870,
In OEIS: - A097728 Chebyshev U(n,x) polynomial evaluated at x=73 = 2*6^2+1.
• Sequence: 1, 147, 21461, 3133159, 457419753, 66780150779,
In OEIS: - A097729 Pell equation solutions (6*a(n))^2 - 37*b(n)^2 = -1 with b(n):=A097730(n), n ≥ 0.
• a(n) = 171a(n-1) - a(n-2).
• Sequence: 1, 170, 29069, 4970629, 849948490, 145336221161,
In OEIS: - A098244 First differences of Chebyshev polynomials S(n,171)=A097844(n) with diophantine property.
• Sequence: 1, 171, 29240, 4999869, 854948359,
In OEIS: - A097844 Chebyshev polynomials S(n,171).
• Sequence: 1, 172, 29411, 5029109, 859948228, 147046117879,
In OEIS: - A097845 Chebyshev polynomials S(n,171) + S(n-1,171) with diophantine property.
• a(n) = 194a(n-1) - a(n-2).
• Sequence: 1, 195, 37829, 7338631, 1423656585,
In OEIS: - A084232 RMS values associated with A084231.
• a(n) = 198a(n-1) - a(n-2).
• Sequence: 1, 197, 39005, 7722793, 1529074009,
In OEIS: - A097733 Pell equation solutions (7*b(n))^2 - 2*(5*a(n))^2 = -1 with b(n):=A097732(n), n ≥ 0. Note that D=50=2*5^2 is not square-free.
• Sequence: 1, 198, 39203, 7761996, 1536836005,
In OEIS: - A097731 Chebyshev U(n,x) polynomial evaluated at x=99 = 2*7^2+1.
• Sequence: 1, 199, 39401, 7801199, 1544598001,
In OEIS: - A097732 Pell equation solutions (7*a(n))^2 - 2*(5*b(n))^2 = -1 with b(n):=A097733(n), n ≥ 0. Note that D=50=2*5^2 is not square-free.
• a(n) = 227a(n-1) - a(n-2).
• Sequence: 1, 226, 51301, 11645101, 2643386626,
In OEIS: - A098247 First differences of Chebyshev polynomials S(n,227)=A098245(n) with diophantine property.
• Sequence: 1, 227, 51528, 11696629, 2655083255,
In OEIS: - A098245 Chebyshev polynomials S(n,227).
• Sequence: 1, 228, 51755, 11748157, 2666779884,
In OEIS: - A098246 Chebyshev polynomials S(n,227) + S(n-1,227) with diophantine property.
• a(n) = 258a(n-1) - a(n-2).
• Sequence: 1, 257, 66305, 17106433, 4413393409,
In OEIS: - A097736 Pell equation solutions (8*b(n))^2 - 65*a(n)^2 = -1 with b(n):=A097735(n), n ≥ 0.
• Sequence: 1, 258, 66563, 17172996, 4430566405,
In OEIS: - A097734 Chebyshev U(n,x) polynomial evaluated at x=129 = 3*43.
• Sequence: 1, 259, 66821, 17239559, 4447739401,
In OEIS: - A097735 Pell equation solutions (8*a(n))^2 - 65*b(n)^2 = -1 with b(n):=A097736(n), n ≥ 0.
• a(n) = 291a(n-1) - a(n-2).
• Sequence: 1, 290, 84389, 24556909, 7145976130,
In OEIS: - A098250 First differences of Chebyshev polynomials S(n,291)=A098248(n) with diophantine property.
• Sequence: 1, 291, 84680, 24641589, 7170617719,
In OEIS: - A098248 Chebyshev polynomials S(n,291).
• Sequence: 1, 292, 84971, 24726269, 7195259308,
In OEIS: - A098249 Chebyshev polynomials S(n,291) + S(n-1,291) with diophantine property.
• a(n) = 322a(n-1) - a(n-2).
• Sequence: 2, 322, 103682, 33385282, 10749957122,
In OEIS: - A089775 Lucas numbers L(12n).
• a(n) = 326a(n-1) - a(n-2).
• Sequence: 1, 325, 105949, 34539049, 11259624025,
In OEIS: - A097739 Pell equation solutions (9*b(n))^2 - 82*a(n)^2 = -1 with b(n):=A097738(n), n ≥ 0.
• Sequence: 1, 326, 106275, 34645324, 11294269349,
In OEIS: - A097737 Chebyshev U(n,x) polynomial evaluated at x=163.
• Sequence: 1, 327, 106601, 34751599, 11328914673,
In OEIS: - A097738 Pell equation solutions (9*a(n))^2 - 82*b(n)^2 = -1 with b(n):=A097739(n), n ≥ 0.
• a(n) = 340a(n-1) - a(n-2).
• Sequence: 1, 170, 57799, 19651490, 6681448801,
In OEIS: - A114048 x such that x^2 - 19*y^2 = 1.
• a(n) = 363a(n-1) - a(n-2).
• Sequence: 1, 362, 131405, 47699653, 17314842634,
In OEIS: - A098253 First differences of Chebyshev polynomials S(n,363)=A098251(n) with diophantine property.
• Sequence: 1, 363, 131768, 47831421, 17362674055,
In OEIS: - A098251 Chebyshev polynomials S(n,363).
• Sequence: 1, 364, 132131, 47963189, 17410505476,
In OEIS: - A098252 Chebyshev polynomials S(n,363) + S(n-1,363) with diophantine property.
• a(n) = 394a(n-1) - a(n-2).
• Sequence: 1, 197, 77617, 30580901, 12048797377,
In OEIS: - A114050 x such that x^2 - 22*y^2 = 1.
• a(n) = 402a(n-1) - a(n-2).
• Sequence: 1, 401, 161201, 64802401, 26050404001,
In OEIS: - A097742 Pell equation solutions (10*b(n))^2 - 101*a(n)^2 = -1 with b(n):=A097741(n), n ≥ 0.
• Sequence: 1, 402, 161603, 64964004, 26115368005,
In OEIS: - A097740 Chebyshev U(n,x) polynomial evaluated at x=201.
• Sequence: 1, 403, 162005, 65125607, 26180332009,
In OEIS: - A097741 Pell equation solutions (10*a(n))^2 - 101*b(n)^2 = -1 with b(n):=A097742(n), n ≥ 0.
• a(n) = 443a(n-1) - a(n-2).
• Sequence: 1, 442, 195805, 86741173, 38426143834,
In OEIS: - A098256 First differences of Chebyshev polynomials S(n,443)=A098254(n) with diophantine property.
• Sequence: 1, 443, 196248, 86937421, 38513081255,
In OEIS: - A098254 Chebyshev polynomials S(n,443).
• Sequence: 1, 444, 196691, 87133669, 38600018676,
In OEIS: - A098255 Chebyshev polynomials S(n,443) + S(n-1,443) with diophantine property.
• a(n) = 486a(n-1) - a(n-2).
• Sequence: 1, 485, 235709, 114554089, 55673051545,
In OEIS: - A097767 Pell equation solutions (11*b(n))^2 - 122*a(n)^2 = -1 with b(n):=A097766(n), n ≥ 0.
• Sequence: 1, 486, 236195, 114790284, 55787841829,
In OEIS: - A097765 Chebyshev U(n,x) polynomial evaluated at x=243=2*11^2+1.
• Sequence: 1, 487, 236681, 115026479, 55902632113,
In OEIS: - A097766 Pell equation solutions (11*a(n))^2 - 122*b(n)^2 = -1 with b(n):=A097767(n), n ≥ 0.
• a(n) = 531a(n-1) - a(n-2).
• Sequence: 1, 530, 281429, 149438269, 79351439410,
In OEIS: - A098259 First differences of Chebyshev polynomials S(n,531)=A098257(n) with diophantine property.
• Sequence: 1, 531, 281960, 149720229, 79501159639,
In OEIS: - A098257 Chebyshev polynomials S(n,531).
• Sequence: 1, 532, 282491, 150002189, 79650879868,
In OEIS: - A098258 Chebyshev polynomials S(n,531) + S(n-1,531) with diophantine property.
• a(n) = 578a(n-1) - a(n-2).
• Sequence: 1, 577, 333505, 192765313, 111418017409,
In OEIS: - A097770 Pell equation solutions (12*b(n))^2 - 145*a(n)^2 = -1 with b(n):=A097769(n), n ≥ 0.
• Sequence: 1, 578, 334083, 193099396, 111611116805,
In OEIS: - A097768 Chebyshev U(n,x) polynomial evaluated at x=289=2*12^2+1.
• Sequence: 1, 579, 334661, 193433479, 111804216201,
In OEIS: - A097769 Pell equation solutions (12*a(n))^2 - 145*b(n)^2 = -1 with b(n):=A097770(n), n ≥ 0.
• a(n) = 627a(n-1) - a(n-2).
• Sequence: 1, 626, 392501, 246097501, 154302740626,
In OEIS: - A098262 First differences of Chebyshev polynomials S(n,627)=A098260(n) with diophantine property.
• Sequence: 1, 627, 393128, 246490629, 154549231255,
In OEIS: - A098260 Chebyshev polynomials S(n,627).
• Sequence: 1, 628, 393755, 246883757, 154795721884,
In OEIS: - A098261 Chebyshev polynomials S(n,627) + S(n-1,627) with diophantine property.
• a(n) = 678a(n-1) - a(n-2).
• Sequence: 1, 677, 459005, 311204713, 210996336409,
In OEIS: - A097773 Pell equation solutions (13*b(n))^2 - 170*a(n)^2 = -1 with b(n):=A097772(n), n ≥ 0.
• Sequence: 1, 678, 459683, 311664396, 211308000805,
In OEIS: - A097771 Chebyshev U(n,x) polynomial evaluated at x=339=2*13^2+1.
• Sequence: 1, 679, 460361, 312124079, 211619665201,
In OEIS: - A097772 Pell equation solutions (13*a(n))^2 - 170*b(n)^2 = -1 with b(n):=A097771(n), n ≥ 0.
• a(n) = 731a(n-1) - a(n-2).
• Sequence: 1, 730, 533629, 390082069, 285149458810,
In OEIS: - A098292 First differences of Chebyshev polynomials S(n,731)=A098263(n) with diophantine property.
• Sequence: 1, 731, 534360, 390616429, 285540075239,
In OEIS: - A098263 Chebyshev polynomials S(n,731).
• Sequence: 1, 732, 535091, 391150789, 285930691668,
In OEIS: - A098291 Chebyshev polynomials S(n,731) + S(n-1,731) with diophantine property.
• a(n) = 786a(n-1) - a(n-2).
• Sequence: 1, 785, 617009, 484968289, 381184458145,
In OEIS: - A097776 Pell equation solutions (14*b(n))^2 - 197*a(n)^2 = -1 with b(n):=A097775(n), n ≥ 0.
• Sequence: 1, 786, 617795, 485586084, 381670044229,
In OEIS: - A097774 Chebyshev U(n,x) polynomial evaluated at x=393=2*14^2+1.
• Sequence: 1, 787, 618581, 486203879, 382155630313,
In OEIS: - A097775 Pell equation solutions (14*a(n))^2 - 197*b(n)^2 = -1 with b(n):=A097776(n), n ≥ 0.
• a(n) = 1298a(n-1) - a(n-2).
• Sequence: 0, 180, 233640, 303264540, 393637139280,
In OEIS: - A075871 Numbers n such that 13*n^2 + 1 is a square.
• Sequence: 1, 649, 842401, 1093435849, 1419278889601,
In OEIS: - A114047 x such that x^2 - 13*y^2 = 1.
• a(n) = 2302a(n-1) - a(n-2).
• Sequence: 1, 1151, 2649601, 6099380351, 14040770918401,
In OEIS: - A114046 x such that x^2 - 92*y^2 = 1.
• a(n) = 2702a(n-1) - a(n-2).
• Sequence: -15, 15, 40545, 109552575, 296011017105,
In OEIS: - A094836 a(n)=2702*a(n-1) - a(n-2); a(-1)=-15; a(0)=15.
• Sequence: 26, 26, 70226, 189750626, 512706121226,
In OEIS: - A094835 a(n) = 2702*a(n-1) - a(n-2); a(-1) = a(0) = 26.

### a(n) = d * a(n-1) + d * a(n-2).

Common properties for sequences with initial terms a(0) = 1 and a(1) = d+1 :

• a(n) equals the number of 00-avoiding words of length n on alphabet {0,1,2, ..., d}. Proof.
• The generating function of this sequence is (1+x)/(1-dx-dx2).

Common properties for sequences with initial terms a(0) = 0, a(1) = 1:

• The generating function of this sequence is x/(1-dx-dx2).
• a(n) = (pn - qn)/(p-q), where p and q are the roots of the equation: x2 - dx - d = 0.

Common properties for sequences with initial terms a(0) = 2, a(1) = d:

• The generating function of this sequence is (2-dx)/(1-dx-dx2).
• a(n) = pn + qn, where p and q are the roots of the equation: x2 - dx - d = 0. Namely a(n) = = ((d+sqrt(d2+4d))/2)n + ((d-sqrt(d2+4d))/2)n. Proof.

Sequences:

• a(n) = - 4a(n-1) - 4a(n-2).
• Sequence: 1, -2, 4, -8, 16, -32, 64, -128, 256,
In OEIS: - A122803 Powers of -2.
• Sequence: 0, -1, 4, -12, 32, -80, 192, -448, 1024,
In OEIS: - A085750 Determinant of the symmetric n X n matrix A defined by A[i,j] = |i-j| for 1 ≤ i,j ≤ n.
• a(n) = - 2a(n-1) - 2a(n-2).
Sequence: 1, -3, 4, -2, -4, 12, -16, 8, 16, -48,
In OEIS: - A078069 Expansion of (1-x)/(1+2*x+2*x^2).
• a(n) = - a(n-1) - a(n-2).
Sequence: 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1,
In OEIS: - A061347 Period 3.
• a(n) = a(n-1) + a(n-2). For d = 1 see a(n) = a(n-1) + a(n-2).
• a(n) = 2a(n-1) + 2a(n-2).
• Sequence: 1, 1, 4, 10, 28, 76, 208, 568, 1552, 4240,
In OEIS: - A026150 a(0) = a(1) = 1; a(n+2) = 2*a(n+1) + 2*a(n).
• Sequence: 1, 2, 6, 16, 44, 120, 328, 896, 2448,
In OEIS: - A002605 a(n+2) = 2*a(n+1) + 2*a(n). Also A080953 a(n)=2(a(n-1)+a(n-2)), a(0)=0, a(1)=1.
• Sequence: 1, 3, 8, 22, 60, 164, 448, 1224, 3344, 9136,
In OEIS: - A028859 a(n+2) = 2a(n+1) + 2a(n).
• Sequence: 2, 2, 8, 20, 56, 152, 416, 1136, 3104, 8480,
In OEIS: - A080040 a(n)=2a(n-1)+2a(n-2), a(0)=2, a(1)=2.
• Sequence: 0, 2, 4, 12, 32, 88, 240, 656, 1792, 4896,
In OEIS: - A106433 Yet another way to compute A028860 ( first two terms different) : 2 X 2 vector Matrix Markov with characteristic Polynomial: x^2-2*x-2.
• Sequence: 0, 3, 6, 18, 48, 132, 360, 984, 2688, 7344,
In OEIS: - A083337 a(n)=2a(n-1)+2a(n-2).
• Sequence: 4, 11, 30, 82, 224, 612, 1672, 4568, 12480,
In OEIS: - A021006 Pisot sequence P(4,11), a(0)=4, a(1)=11, a(n+1) is the nearest integer to a(n)^2/a(n-1). Evidently satisfies a(n)=2a(n-1)+2a(n-2).
• Sequence: 0, 4, 8, 24, 64, 176, 480, 1312, 3584,
In OEIS: - A116556 a(0)=0, a(1)=4, a(n)=2a(n-1)+2a(n-2)
• a(n) = 3a(n-1) + 3a(n-2).
• Sequence: 1, 3, 12, 45, 171, 648, 2457,
In OEIS: - A030195 a(n) = 3*a(n-1)+3*a(n-2), a(0)=0, a(1)=1.
• Sequence: 1, 4, 15, 57, 216, 819,
In OEIS: - A125145 a(n) = 3a(n-1) + 3a(n-2). a(0) = 1, a(1) = 4.
• Sequence: 0, 3, 9, 36, 135, 513, 1944,
In OEIS: - A106435 a(n) = 3*a(n-1)+3*a(n-2), a(0)=0, a(1)=3.
• Sequence: 3, 15, 54, 207, 783, 2970, 11259,
In OEIS: - A085480 a(n) = p^n + q^n, where p = (3 + sqrt 21)/2, q = (3 - sqrt 21)/2.
• Sequence: 1, 6, 21, 81, 306, 1161, 4401, 16686,
In OEIS: - A108306 Expansion of (3*x+1)/(1-3*x-3*x^2).
• a(n) = 4a(n-1) + 4a(n-2).
• Sequence: 1, 4, 20, 96, 464, 2240, 10816, 52224,
In OEIS: - A057087 Scaled Chebyshev U-polynomials evaluated at i. Generalized Fibonacci sequence.
• Sequence: 1, 5, 24, 116, 560, 2704, 13056, 63040,
In OEIS: - A086347 On a 3 X 3 board, number of n-move routes of chess king ending at a given side cell.
• Sequence: 1, 2, 12, 56, 272, 1312, 6336, 30592,
In OEIS: - A084128 Generalized Fibonacci sequence.
• Sequence: 1, 0, 4, 16, 80, 384, 1856, 8960, 43264,
In OEIS: - A094013 Expansion of (1-4x)/(1-4x-4x^2). Also A106568 First entry of the vector (M^n)v, where M is the 2x2 matrix [[0,4],[1,4]] and v is the column vector [0,1].
• a(n) = 5a(n-1) + 5a(n-2).
• Sequence: 1, 5, 30, 175, 1025, 6000, 35125, 205625,
In OEIS: - A057088 Scaled Chebyshev U-polynomials evaluated at i*sqrt(5)/2. Generalized Fibonacci sequence.
• Sequence: 0, 5, 25, 150, 875, 5125, 30000, 175625,
In OEIS: - A106565 Let M={{0, 5}, {1, 5}}, v[n]=M.v[n-1]; then a(n) =v[n][[1]].
• a(n) = 6a(n-1) + 6a(n-2).
• Sequence: 1, 6, 42, 288, 1980, 13608, 93528, 642816,
In OEIS: - A057089 Scaled Chebyshev U-polynomials evaluated at i*sqrt(6)/2. Generalized Fibonacci sequence.
• Sequence: 8, 55, 378, 2598, 17856, 122724, 843480,
In OEIS: - A010924 Pisot sequence E(8,55), a(n)=[ a(n-1)^2/a(n-2)+1/2 ].
• a(n) = 7a(n-1) + 7a(n-2).
Sequence: 1, 7, 56, 441, 3479, 27440, 216433,
In OEIS: - A057090 Scaled Chebyshev U-polynomials evaluated at i*sqrt(7)/2. Generalized Fibonacci sequence.
• a(n) = 8a(n-1) + 8a(n-2).
Sequence: 1, 8, 72, 640, 5696, 50688, 451072,
In OEIS: - A057091 Scaled Chebyshev U-polynomials evaluated at i*sqrt(2). Generalized Fibonacci sequence.
• a(n) = 9a(n-1) + 9a(n-2).
Sequence: 1, 9, 90, 891, 8829, 87480, 866781,
In OEIS: - A057092 Scaled Chebyshev U-polynomials evaluated at i*3/2. Generalized Fibonacci sequence.
• a(n) = 10a(n-1) + 10a(n-2).
Sequence: 1, 10, 110, 1200, 13100, 143000, 1561000,
In OEIS: - A057093 Scaled Chebyshev U-polynomials evaluated at i*sqrt(10)/2. Generalized Fibonacci sequence.

### a(n) = a(n-1) + a(n-2).

Common properties for these sequences:

• The sum of the first n terms: a(1) + a(2) + ... + a(n-1) + a(n) equals a(n+2) - a(2).
• The sum of the first n odd numbered terms: a(1) + a(3) + ... + a(2n-3) + a(2n-1) equals a(2n) - a(2) + a(1).
• The sum of the first n even numbered terms: a(2) + a(4) + ... + a(2n-2) + a(2n) equals a(2n+1) - a(1).

Properties for the sequence with initial terms a(0) = 1 and a(1) = 2 (shifted Fibonacci sequence):

• a(n) equals the number of 00-avoiding words of length n on alphabet {0,1}.
• The generating function of this sequence is (1+x)/(1-x-x2).

Properties for the sequence with initial terms a(0) = 0, a(1) = 1 (Fibonacci sequence):

• The generating function of this sequence is x/(1-x-x2).
• a(n) = (pn - qn)/(p-q), where p and q are the roots of the equation: x2 - x - 1 = 0. Namely a(n) = (φn + (-1/φ)n)/sqrt(5), where φ is the golden ratio.
• Asymptotically a(n) = Round(φn/sqrt(5)), where φ is the golden ratio.

Properties for the sequence with initial terms a(0) = 2, a(1) = 1 (Lucas sequence):

• The generating function of this sequence is (2-x)/(1-x-x2).
• a(n) = pn + qn, where p and q are the roots of the equation: x2 - x - 1 = 0. Namely a(n) = ((1+sqrt(5))/2)n + ((1-sqrt(5))/2)n = φn + (-1/φ)n, where φ is the golden ratio.
• Asymptotically a(n) = Round(φn), where φ is the golden ratio.

Sequences:

• a(n) = a(n-1) + a(n-2).
• Sequence: 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233,
In OEIS: - A000045 Fibonacci numbers: F(n) = F(n-1) + F(n-2), F(0) = 0, F(1) = 1. Also A020695 Pisot sequence E(2,3). Also A020701 Pisot sequences E(3,5), P(3,5). Also A020712 Pisot sequences E(5,8), P(5,8).
• Sequence: 1, 3, 4, 7, 11, 18, 29, 47, 76, 123,
In OEIS: - A000032 Lucas numbers (beginning at 2): L(n) = L(n-1) + L(n-2); and A000204 Lucas numbers (beginning with 1): L(n) = L(n-1) + L(n-2) with L(1) = 1, L(2) = 3.
• Sequence: 1, 4, 5, 9, 14, 23, 37, 60, 97, 157,
In OEIS: - A000285 a(n) = a(n-1) + a(n-2). Also A104449 Fibonacci-type sequence. Each term is the sum of the two previous terms.
• Sequence: 1, 5, 6, 11, 17, 28, 45, 73, 118, 191,
In OEIS: - A022095 Fibonacci sequence beginning 1 5.
• Sequence: 2, 4, 6, 10, 16, 26, 42, 68, 110, 178,
In OEIS: - A090991 Number of meaningful differential operations of the n-th order on the space R^6. Also A118658 L_n - F_n where L_n is the Lucas Number and F_n is the Fibonacci Number. Also A078642 Numbers with two representations as the sum of two Fibonacci numbers.
• Sequence: 2, 5, 7, 12, 19, 31, 50, 81, 131, 212,
In OEIS: - A001060 a(n) = a(n-1) + a(n-2); and A013655 F(n)+L(n), where F(n) and L(n) are Fibonacci and Lucas numbers respectively.
• Sequence: 2, 6, 8, 14, 22, 36, 58, 94, 152, 246,
In OEIS: - A022112 Fibonacci sequence beginning 2 6.
• Sequence: 2, 7, 9, 16, 25, 41, 66, 107, 173, 280,
In OEIS: - A022113 Fibonacci sequence beginning 2 7.
• Sequence: 2, 9, 11, 20, 31, 51, 82, 133, 215,
In OEIS: - A022114 Fibonacci sequence beginning 2 9.
• Sequence: 2, 10, 12, 22, 34, 56, 90, 146, 236,
In OEIS: - A022367 Fibonacci sequence beginning 2 10.
• Sequence: 2, 11, 13, 24, 37, 61, 98, 159, 257,
In OEIS: - A022115 Fibonacci sequence beginning 2 11.
• Sequence: 2, 12, 14, 26, 40, 66, 106, 172, 278,
In OEIS: - A022368 Fibonacci sequence beginning 2 12.
• Sequence: 2, 13, 15, 28, 43, 71, 114, 185, 299,
In OEIS: - A022116 Fibonacci sequence beginning 2 13.
• Sequence: 2, 14, 16, 30, 46, 76, 122, 198, 320,
In OEIS: - A022369 Fibonacci sequence beginning 2 14.
• Sequence: 2, 15, 17, 32, 49, 81, 130, 211, 341,
In OEIS: - A022117 Fibonacci sequence beginning 2 15.
• Sequence: 2, 16, 18, 34, 52, 86, 138, 224, 362,
In OEIS: - A022370 Fibonacci sequence beginning 2 16.
• Sequence: 2, 17, 19, 36, 55, 91, 146, 237, 383,
In OEIS: - A022118 Fibonacci sequence beginning 2 17.
• Sequence: 2, 18, 20, 38, 58, 96, 154, 250, 404,
In OEIS: - A022371 Fibonacci sequence beginning 2 18.
• Sequence: 2, 19, 21, 40, 61, 101, 162, 263, 425,
In OEIS: - A022119 Fibonacci sequence beginning 2 19.
• Sequence: 2, 20, 22, 42, 64, 106, 170, 276, 446,
In OEIS: - A022372 Fibonacci sequence beginning 2 20.
• Sequence: 2, 22, 24, 46, 70, 116, 186, 302, 488,
In OEIS: - A022373 Fibonacci sequence beginning 2 22.
• Sequence: 2, 24, 26, 50, 76, 126, 202, 328, 530,
In OEIS: - A022374 Fibonacci sequence beginning 2 24.
• Sequence: 2, 26, 28, 54, 82, 136, 218, 354, 572,
In OEIS: - A022375 Fibonacci sequence beginning 2 26.
• Sequence: 2, 28, 30, 58, 88, 146, 234, 380, 614,
In OEIS: - A022376 Fibonacci sequence beginning 2 28.
• Sequence: 2, 30, 32, 62, 94, 156, 250, 406, 656,
In OEIS: - A022377 Fibonacci sequence beginning 2 30.
• Sequence: 2, 32, 34, 66, 100, 166, 266, 432, 698,
In OEIS: - A022378 Fibonacci sequence beginning 2 32.
• Sequence: 0, 3, 3, 6, 9, 15, 24, 39, 63, 102,
In OEIS: - A022086 Fibonacci sequence beginning 0 3.
• Sequence: 3, 7, 10, 17, 27, 44, 71, 115, 186, 301,
In OEIS: - A022120 Fibonacci sequence beginning 3 7.
• Sequence: 3, 8, 11, 19, 30, 49, 79, 128, 207, 335,
In OEIS: - A022121 Fibonacci sequence beginning 3 8.
• Sequence: 3, 9, 12, 21, 33, 54, 87, 141, 228, 369,
In OEIS: - A022379 Fibonacci sequence beginning 3 9.
• Sequence: 3, 10, 13, 23, 36, 59, 95, 154, 249, 403,
In OEIS: - A022122 Fibonacci sequence beginning 3 10.
• Sequence: 3, 11, 14, 25, 39, 64, 103, 167, 270, 437,
In OEIS: - A022123 Fibonacci sequence beginning 3 11.
• Sequence: 3, 12, 15, 27, 42, 69, 111, 180, 291, 471,
In OEIS: - A022380 Fibonacci sequence beginning 3 12.
• Sequence: 3, 13, 16, 29, 45, 74, 119, 193, 312, 505,
In OEIS: - A022124 Fibonacci sequence beginning 3 13.
• Sequence: 3, 14, 17, 31, 48, 79, 127, 206, 333, 539,
In OEIS: - A022125 Fibonacci sequence beginning 3 14.
• Sequence: 3, 15, 18, 33, 51, 84, 135, 219, 354, 573,
In OEIS: - A022381 Fibonacci sequence beginning 3 15.
• Sequence: 3, 16, 19, 35, 54, 89, 143, 232, 375, 607,
In OEIS: - A022126 Fibonacci sequence beginning 3 16.
• Sequence: 3, 17, 20, 37, 57, 94, 151, 245, 396, 641,
In OEIS: - A022127 Fibonacci sequence beginning 3 17.
• Sequence: 3, 19, 22, 41, 63, 104, 167, 271, 438, 709,
In OEIS: - A022128 Fibonacci sequence beginning 3 19.
• Sequence: 3, 20, 23, 43, 66, 109, 175, 284, 459, 743,
In OEIS: - A022129 Fibonacci sequence beginning 3 20.
• Sequence: 0, 4, 4, 8, 12, 20, 32, 52, 84, 136, 220,
In OEIS: - A022087 Fibonacci sequence beginning 0 4.
• Sequence: 4, 9, 13, 22, 35, 57, 92, 149, 241, 390,
In OEIS: - A022130 Fibonacci sequence beginning 4 9.
• Sequence: 4, 10, 14, 24, 38, 62, 100, 162, 262, 424,
In OEIS: - A022382 Fibonacci sequence beginning 4 10.
• Sequence: 4, 11, 15, 26, 41, 67, 108, 175, 283, 458,
In OEIS: - A022131 Fibonacci sequence beginning 4 11.
• Sequence: 4, 13, 17, 30, 47, 77, 124, 201, 325, 526,
In OEIS: - A022132 Fibonacci sequence beginning 4 13.
• Sequence: 4, 14, 18, 32, 50, 82, 132, 214, 346, 560,
In OEIS: - A022383 Fibonacci sequence beginning 4 14.
• Sequence: 4, 15, 19, 34, 53, 87, 140, 227, 367, 594,
In OEIS: - A022133 Fibonacci sequence beginning 4 15.
• Sequence: 4, 17, 21, 38, 59, 97, 156, 253, 409, 662,
In OEIS: - A022134 Fibonacci sequence beginning 4 17.
• Sequence: 4, 18, 22, 40, 62, 102, 164, 266, 430, 696,
In OEIS: - A022384 Fibonacci sequence beginning 4 18.
• Sequence: 4, 19, 23, 42, 65, 107, 172, 279, 451, 730,
In OEIS: - A022135 Fibonacci sequence beginning 4 19.
• Sequence: 4, 22, 26, 48, 74, 122, 196, 318, 514, 832,
In OEIS: - A022385 Fibonacci sequence beginning 4 22.
• Sequence: 4, 26, 30, 56, 86, 142, 228, 370, 598, 968,
In OEIS: - A022386 Fibonacci sequence beginning 4 26.
• Sequence: 4, 30, 34, 64, 98, 162, 260, 422, 682, 1104,
In OEIS: - A022387 Fibonacci sequence beginning 4 30.
• Sequence: 0, 5, 5, 10, 15, 25, 40, 65, 105, 170, 275,
In OEIS: - A022088 Fibonacci sequence beginning 0 5.
• Sequence: 5, 11, 16, 27, 43, 70, 113, 183, 296, 479,
In OEIS: - A022136 Fibonacci sequence beginning 5 11.
• Sequence: 5, 12, 17, 29, 46, 75, 121, 196, 317, 513,
In OEIS: - A022137 Fibonacci sequence beginning 5 12.
• Sequence: 5, 13, 18, 31, 49, 80, 129, 209, 338, 547,
In OEIS: - A022138 Fibonacci sequence beginning 5 13.
• Sequence: 5, 14, 19, 33, 52, 85, 137, 222, 359, 581,
In OEIS: - A022139 Fibonacci sequence beginning 5 14.
• Sequence: 5, 16, 21, 37, 58, 95, 153, 248, 401, 649,
In OEIS: - A022140 Fibonacci sequence beginning 5 16.
• Sequence: 5, 17, 22, 39, 61, 100, 161, 261, 422, 683,
In OEIS: - A022141 Fibonacci sequence beginning 5 17.
• Sequence: 5, 18, 23, 41, 64, 105, 169, 274, 443, 717,
In OEIS: - A022142 Fibonacci sequence beginning 5 18.
• Sequence: 5, 19, 24, 43, 67, 110, 177, 287, 464, 751,
In OEIS: - A022143 Fibonacci sequence beginning 5 19.
• Sequence: 5, 19, 24, 43, 67, 110, 177, 287, 464, 751,
In OEIS: - A022143 Fibonacci sequence beginning 5 19.
• Sequence: 0, 6, 6, 12, 18, 30, 48, 78, 126, 204, 330,
In OEIS: - A022089 Fibonacci sequence beginning 0 6.
• Sequence: 1, 6, 7, 13, 20, 33, 53, 86, 139, 225, 364,
In OEIS: - A022096 Fibonacci sequence beginning 1 6.
• Sequence: 6, 13, 19, 32, 51, 83, 134, 217, 351, 568,
In OEIS: - A022388 Fibonacci sequence beginning 6 13.
• Sequence: 0, 7, 7, 14, 21, 35, 56, 91, 147, 238, 385,
In OEIS: - A022090 Fibonacci sequence beginning 0 7.
• Sequence: 1, 7, 8, 15, 23, 38, 61, 99, 160, 259, 419,
In OEIS: - A022097 Fibonacci sequence beginning 1 7.
• Sequence: 7, 15, 22, 37, 59, 96, 155, 251, 406, 657,
In OEIS: - A022389 Fibonacci sequence beginning 7 15.
• Sequence: 7, 26, 33, 59, 92, 151, 243, 394, 637, 1031,
In OEIS: - A098127 Fibonacci sequence with a(1)=7 and a(2) = 26.
• Sequence: 0, 8, 8, 16, 24, 40, 64, 104, 168, 272, 440,
In OEIS: - A022091 Fibonacci sequence beginning 0 8.
• Sequence: 1, 8, 9, 17, 26, 43, 69, 112, 181, 293, 474,
In OEIS: - A022098 Fibonacci sequence beginning 1 8.
• Sequence: 8, 17, 25, 42, 67, 109, 176, 285, 461, 746,
In OEIS: - A022390 Fibonacci sequence beginning 8 17.
• Sequence: 0, 9, 9, 18, 27, 45, 72, 117, 189, 306, 495,
In OEIS: - A022092 Fibonacci sequence beginning 0 9.
• Sequence: 1, 9, 10, 19, 29, 48, 77, 125, 202, 327, 529,
In OEIS: - A022099 Fibonacci sequence beginning 1 9.
• Sequence: 0, 10, 10, 20, 30, 50, 80, 130, 210, 340,
In OEIS: - A022093 Fibonacci sequence beginning 0 10.
• Sequence: 1, 10, 11, 21, 32, 53, 85, 138, 223, 361,
In OEIS: - A022100 Fibonacci sequence beginning 1 10.
• Sequence: 0, 11, 11, 22, 33, 55, 88, 143, 231, 374,
In OEIS: - A022345 Fibonacci sequence beginning 0 11.
• Sequence: 1, 11, 12, 23, 35, 58, 93, 151, 244, 395,
In OEIS: - A022101 Fibonacci sequence beginning 1 11.
• Sequence: 11, 23, 34, 57, 91, 148, 239, 387, 626, 1013,
In OEIS: - A097657 Fibonacci sequence with first two terms 11 and 23.
• Sequence: 0, 12, 12, 24, 36, 60, 96, 156, 252, 408,
In OEIS: - A022346 Fibonacci sequence beginning 0 12.
• Sequence: 1, 12, 13, 25, 38, 63, 101, 164, 265, 429,
In OEIS: - A022102 Fibonacci sequence beginning 1 12.
• Sequence: 12, 67, 79, 146, 225, 371, 596, 967, 1563,
In OEIS: - A091074 Fibonacci-like sequence beginning (12, 67).
• Sequence: 0, 13, 13, 26, 39, 65, 104, 169, 273, 442,
In OEIS: - A022347 Fibonacci sequence beginning 0 13.
• Sequence: 1, 13, 14, 27, 41, 68, 109, 177, 286, 463,
In OEIS: - A022103 Fibonacci sequence beginning 1 13.
• Sequence: 0, 14, 14, 28, 42, 70, 112, 182, 294, 476,
In OEIS: - A022348 Fibonacci sequence beginning 0 14.
• Sequence: 1, 14, 15, 29, 44, 73, 117, 190, 307, 497,
In OEIS: - A022104 Fibonacci sequence beginning 1 14.
• Sequence: 0, 15, 15, 30, 45, 75, 120, 195, 315, 510,
In OEIS: - A022349 Fibonacci sequence beginning 0 15.
• Sequence: 1, 15, 16, 31, 47, 78, 125, 203, 328, 531,
In OEIS: - A022105 Fibonacci sequence beginning 1 15.
• Sequence: 0, 16, 16, 32, 48, 80, 128, 208, 336, 544,
In OEIS: - A022350 Fibonacci sequence beginning 0 16.
• Sequence: 1, 16, 17, 33, 50, 83, 133, 216, 349, 565,
In OEIS: - A022106 Fibonacci sequence beginning 1 16.
• Sequence: 0, 17, 17, 34, 51, 85, 136, 221, 357, 578,
In OEIS: - A022351 Fibonacci sequence beginning 0 17.
• Sequence: 1, 17, 18, 35, 53, 88, 141, 229, 370, 599,
In OEIS: - A022107 Fibonacci sequence beginning 1 17.
• Sequence: 0, 18, 18, 36, 54, 90, 144, 234, 378, 612,
In OEIS: - A022352 Fibonacci sequence beginning 0 18.
• Sequence: 1, 18, 19, 37, 56, 93, 149, 242, 391, 633,
In OEIS: - A022108 Fibonacci sequence beginning 1 18.
• Sequence: 0, 19, 19, 38, 57, 95, 152, 247, 399, 646,
In OEIS: - A022353 Fibonacci sequence beginning 0 19.
• Sequence: 1, 19, 20, 39, 59, 98, 157, 255, 412, 667,
In OEIS: - A022109 Fibonacci sequence beginning 1 19.
• Sequence: 0, 20, 20, 40, 60, 100, 160, 260, 420, 680,
In OEIS: - A022354 Fibonacci sequence beginning 0 20.
• Sequence: 1, 20, 21, 41, 62, 103, 165, 268, 433, 701,
In OEIS: - A022110 Fibonacci sequence beginning 1 20.
• Sequence: 0, 21, 21, 42, 63, 105, 168, 273, 441, 714,
In OEIS: - A022355 Fibonacci sequence beginning 0 21.
• Sequence: 1, 21, 22, 43, 65, 108, 173, 281, 454, 735,
In OEIS: - A022391 Fibonacci sequence beginning 1 21.
• Sequence: 0, 22, 22, 44, 66, 110, 176, 286, 462, 748,
In OEIS: - A022356 Fibonacci sequence beginning 0 22.
• Sequence: 1, 22, 23, 45, 68, 113, 181, 294, 475, 769,
In OEIS: - A022392 Fibonacci sequence beginning 1 22.
• Sequence: 0, 23, 23, 46, 69, 115, 184, 299, 483, 782,
In OEIS: - A022357 Fibonacci sequence beginning 0 23.
• Sequence: 1, 23, 24, 47, 71, 118, 189, 307, 496, 803,
In OEIS: - A022393 Fibonacci sequence beginning 1 23.
• Sequence: 0, 24, 24, 48, 72, 120, 192, 312, 504, 816,
In OEIS: - A022358 Fibonacci sequence beginning 0 24.
• Sequence: 1, 24, 25, 49, 74, 123, 197, 320, 517, 837,
In OEIS: - A022394 Fibonacci sequence beginning 1 24.
• Sequence: 0, 25, 25, 50, 75, 125, 200, 325, 525, 850,
In OEIS: - A022359 Fibonacci sequence beginning 0 25.
• Sequence: 1, 25, 26, 51, 77, 128, 205, 333, 538, 871,
In OEIS: - A022395 Fibonacci sequence beginning 1 25.
• Sequence: 0, 26, 26, 52, 78, 130, 208, 338, 546, 884,
In OEIS: - A022360 Fibonacci sequence beginning 0 26.
• Sequence: 1, 26, 27, 53, 80, 133, 213, 346, 559, 905,
In OEIS: - A022396 Fibonacci sequence beginning 1 26.
• Sequence: 0, 27, 27, 54, 81, 135, 216, 351, 567, 918,
In OEIS: - A022361 Fibonacci sequence beginning 0 27.
• Sequence: 1, 27, 28, 55, 83, 138, 221, 359, 580, 939,
In OEIS: - A022397 Fibonacci sequence beginning 1 27.
• Sequence: 0, 28, 28, 56, 84, 140, 224, 364, 588, 952,
In OEIS: - A022362 Fibonacci sequence beginning 0 28.
• Sequence: 1, 28, 29, 57, 86, 143, 229, 372, 601, 973,
In OEIS: - A022398 Fibonacci sequence beginning 1 28.
• Sequence: 0, 29, 29, 58, 87, 145, 232, 377, 609, 986,
In OEIS: - A022363 Fibonacci sequence beginning 0 29.
• Sequence: 1, 29, 30, 59, 89, 148, 237, 385, 622,
In OEIS: - A022399 Fibonacci sequence beginning 1 29.
• Sequence: 0, 30, 30, 60, 90, 150, 240, 390, 630, 1020,
In OEIS: - A022364 Fibonacci sequence beginning 0 30.
• Sequence: 1, 30, 31, 61, 92, 153, 245, 398, 643, 1041,
In OEIS: - A022400 Fibonacci sequence beginning 1 30.
• Sequence: 0, 31, 31, 62, 93, 155, 248, 403, 651, 1054,
In OEIS: - A022365 Fibonacci sequence beginning 0 31.
• Sequence: 1, 31, 32, 63, 95, 158, 253, 411, 664, 1075,
In OEIS: - A022401 Fibonacci sequence beginning 1 31.
• Sequence: 0, 32, 32, 64, 96, 160, 256, 416, 672, 1088,
In OEIS: - A022366 Fibonacci sequence beginning 0 32.
• Sequence: 1, 32, 33, 65, 98, 163, 261, 424, 685, 1109,
In OEIS: - A022402 Fibonacci sequence beginning 1 32.
• Sequence: 110, 211, 321, 532, 853, 1385, 2238, 3623,
In OEIS: - A120727 a(n) = a(n-1) + a(n-2), starting with 110, 211.
• Sequence: 407389224418, 76343678551, 483732902969,
In OEIS: - A082411 Second-order linear recurrence sequence with a(n) = a(n-1) + a(n-2).
• Sequence: 20615674205555510, 3794765361567513, 24410439567123023,
In OEIS: - A083216 Second-order linear recurrence sequence with a(n) = a(n-1) + a(n-2).
• Sequence: 62638280004239857, 49463435743205655, 112101715747445512,
In OEIS: - A083105 Second-order linear recurrence sequence with a(n) = a(n-1) + a(n-2).
• Sequence: 331635635998274737472200656430763, 1510028911088401971189590305498785,
In OEIS: - A083104 Second-order linear recurrence sequence with a(n) = a(n-1) + a(n-2).
• Sequence: 31786772701928802632268715130455793, 1059683225053915111058165141686995,
In OEIS: - A083103 Second-order linear recurrence sequence with a(n) = a(n-1) + a(n-2).

### a(n) = d * a(n-1) + a(n-2).

Common properties for sequences starting 1, d+1:

• The generating function of this sequence is (1+x)/(1-dx-x2).
• a(n) is the number of words in the alphabet {0, 1, 2, ..., d} avoiding "11", "22", ..., "dd". Proof.

Common properties for sequences starting 2, d:

• The generating function of this sequence is (2-d)/(1-dx-x2).
• a(n) = pn + qn, where p and q are the roots of the equation: x2 - dx - 1 = 0. Namely a(n) = ((d+sqrt(d2+4))/2)n + ((d-sqrt(d2+4))/2)n. Proof.
• Asymptotically a(n) = Round(pn), where p is the largest root of the equation: x2 - dx - 1 = 0. That is a(n) approaches ((d+sqrt(d2+4))/2)n.

Common properties for sequences with positive d = 2k.

• Sequence starting with 1 and k:
• The generating function of this sequence is (1-k)/(1-2kx-x2).
• a(n) = (pn + qn)/2, where p and q are the roots of the equation: x2 - 2kx - 1 = 0. (Namely p = k + sqrt(k2+1) and q = k - sqrt(k2+1)).
• a(n) are numerators of continued fractions converging to the square root of k2+1. Proof.
• Sequence starting with 1 and 2k:
• The generating function of this sequence is 1/(1-2kx-x2).
• a(n) are denominators of continued fractions converging to the square root of k2+1. Proof.

Sequences:

• a(n) = -11a(n-1) + a(n-2).
• Sequence: 1, 1, -10, 111, -1231, 13652, -151403,
In OEIS: - A122574 a(1)=a(2)=1, a(n)=-11a(n-1)+a(n-2)
• a(n) = -4a(n-1) + a(n-2).
• Sequence: 1, -5, 21, -89, 377, -1597, 6765, -28657,
In OEIS: - A099843 A transform of the Fibonacci numbers.
• a(n) = -2a(n-1) + a(n-2).
• Sequence: 1, -2, 5, -12, 29, -70, 169, -408, 985,
In OEIS: - A077985 Expansion of 1/(1+2*x-x^2).
• Sequence: 1, -3, 4, -7, 11, -18, 29, -47, 76, -123,
In OEIS: - A075193 "Inverted" Lucas numbers (see Comments).
• a(n) = -a(n-1) + a(n-2).
• Sequence: 1, 2, -1, 3, -4, 7, -11, 18, -29, 47,
In OEIS: - A061084 Fibonacci-type sequence based on subtraction: a(0) = 1, a(1) = 2 and a(n) = a(n-2)-a(n-1).
• a(n) = a(n-2). For d = 0 see TBD.
• a(n) = a(n-1) + a(n-2). For d = 1 see a(n) = a(n-1) + a(n-2).
• a(n) = 2a(n-1) + a(n-2).
• Sequence: 1, 2, 5, 12, 29, 70, 169, 408, 985,
In OEIS: - A000129 Pell numbers: a(0) = 0, a(1) = 1; for n > 1, a(n) = 2*a(n-1) + a(n-2). Also A069306 Number of 2 X n binary arrays with a path of adjacent 1's from upper left corner to anywhere in right hand column.
• Sequence: 1, 3, 7, 17, 41, 99, 239, 577, 1393,
In OEIS: - A001333 Numerators of continued fraction convergents to sqrt(2). Also A078057 Expansion of (1+x)/(1-2*x-x^2).
• Sequence: 1, 4, 9, 22, 53, 128, 309, 746, 1801,
In OEIS: - A048654 Generalized Pellian with second term equal to 4.
• Sequence: 1, 5, 11, 27, 65, 157, 379, 915, 2209,
In OEIS: - A048655 Generalized Pellian with second term equal to 5.
• Sequence: 1, 6, 13, 32, 77, 186, 449, 1084, 2617,
In OEIS: - A048693 Generalized Pellian with 2nd term equal to 6.
• Sequence: 1, 7, 15, 37, 89, 215, 519, 1253, 3025,
In OEIS: - A048694 Generalized Pellian with second term equal to 7.
• Sequence: 1, 8, 17, 42, 101, 244, 589, 1422, 3433,
In OEIS: - A048695 Generalized Pellian with second term equal to 8.
• Sequence: 1, 9, 19, 47, 113, 273, 659, 1591, 3841,
In OEIS: - A048696 Generalized Pellian with second term equal to 9.
• Sequence: 1, 10, 21, 52, 125, 302, 729, 1760, 4249,
In OEIS: - A048697 Generalized Pellian with second term equal to 10.
• Sequence: 2, 2, 6, 14, 34, 82, 198, 478, 1154,
In OEIS: - A002203 Companion Pell numbers: a(n) = 2a(n-1) + a(n-2).
• Sequence: 2, 3, 8, 19, 46, 111, 268, 647, 1562,
In OEIS: - A078343 a(0) = -1, a(1) = 2; a(n) = 2*a(n-1) + a(n-2).
• Sequence: 5, 14, 33, 80, 193, 466, 1125, 2716,
In OEIS: - A105082 G.f. (5+4x)/(1-2x-x^2).
• a(n) = 3a(n-1) + a(n-2).
• Sequence: 1, 2, 7, 23, 76, 251, 829, 2738, 9043,
In OEIS: - A052924 G.f.: (1-x)/(1-3*x-x^2).
• Sequence: 1, 3, 10, 33, 109, 360, 1189, 3927,
In OEIS: - A006190 a(n) = 3*a(n-1) + a(n-2). Also A020704 Pisot sequences E(3,10), P(3,10).
• Sequence: 1, 4, 13, 43, 142, 469, 1549, 5116,
In OEIS: - A003688 a(n) = 3*a(n-1) + a(n-2).
• Sequence: 1, 5, 16, 53, 175, 578, 1909, 6305,
In OEIS: - A006190 a(n) = A108300 a(n+2) = 3*a(n+1) + a(n), a(0) = 1, a(1) = 5.
• Sequence: 2, 3, 11, 36, 119, 393, 1298, 4287,
In OEIS: - A006497 a(n) = 3a(n-1) + a(n-2).
• Sequence: 2, 7, 30, 127, 538, 2279, 9654, 40895,
In OEIS: - A097924 Sequence relates numerators and denominators in the continued fraction convergents to sqrt(5).
• a(n) = 4a(n-1) + a(n-2).
• Sequence: 1, 2, 9, 38, 161, 682, 2889,
In OEIS: - A001077 Numerators of continued fraction convergents to sqrt(5).
• Sequence: 1, 3, 13, 55, 233, 987, 4181,
In OEIS: - A033887 Fibonacci(3n+1).
• Sequence: 1, 4, 17, 72, 305, 1292, 5473,
In OEIS: - A001076 Denominators of continued fraction convergents to sqrt(5).
• Sequence: 1, 5, 21, 89, 377, 1597, 6765,
In OEIS: - A015448 Generalized Fibonacci numbers: a(n) = 4*a(n-1) + a(n-2).
• Sequence: 1, 6, 25, 106, 449, 1902, 8057,
In OEIS: - A048875 Generalized Pellian with second term of 6.
• Sequence: 1, 7, 29, 123, 521, 2207, 9349,
In OEIS: - A048876 Generalized Pellian with second term of 7.
• Sequence: 1, 8, 33, 140, 593, 2512, 10641,
In OEIS: - A048877 Generalized Pellian with second term of 8.
• Sequence: 1, 9, 37, 157, 665, 2817, 11933,
In OEIS: - A048878 Generalized Pellian with second term of 9.
• Sequence: 1, 10, 41, 174, 737, 3122, 13225,
In OEIS: - A048879 Generalized Pellian with second term of 10.
• Sequence: 2, 4, 18, 76, 322, 1364, 5778,
In OEIS: - A014448 Even Lucas numbers: L(3n).
• Sequence: 2, 8, 34, 144, 610, 2584, 10946,
In OEIS: - A014445 Even Fibonacci numbers; or, Fibonacci_{3k}.
• a(n) = 5a(n-1) + a(n-2).
• Sequence: 1, 4, 21, 109, 566, 2939, 15261, 79244,
In OEIS: - A100237 Secondary diagonal of triangle A100235 divided by row number: a(n) = A100235(n+1,n)/(n+1) for n>=0.
• Sequence: 1, 5, 26, 135, 701, 3640, 18901, 98145,
In OEIS: - A052918 a(0)=1, a(1)=5, a(n+1) = 5*a(n) + a(n-1).
• Sequence: 1, 6, 31, 161, 836, 4341, 22541,
In OEIS: - A015449 Generalized Fibonacci numbers.
• Sequence: 2, 5, 27, 140, 727, 3775, 19602, 101785,
In OEIS: - A087130 a(n)=5*a(n-1)+a(n-2); a(0)=2, a(1)=5.
• a(n) = 6a(n-1) + a(n-2).
• Sequence: 1, 1, 7, 43, 265, 1633, 10063,
In OEIS: - A015451 a(n) = 6 a(n-1) + a(n-2).
• Sequence: 1, 3, 19, 117, 721, 4443, 27379,
In OEIS: - A005667 Numerators of continued fraction convergents to sqrt(10).
• Sequence: 1, 6, 37, 228, 1405, 8658, 53353,
In OEIS: - A005668 Denominators of continued fraction convergents to sqrt(10).
• Sequence: 2, 6, 38, 234, 1442, 8886, 54758,
In OEIS: - A085447 a(n) = 6*a(n-1) + a(n-2), starting with a(0)=2 and a(1)=6.
• Sequence: 0, 2, 12, 74, 456, 2810, 17316,
In OEIS: - A078469 Number of different compositions of the ladder graph L_n.
• a(n) = 7a(n-1) + a(n-2).
• Sequence: 1, 1, 8, 57, 407, 2906, 20749, 148149,
In OEIS: - A015453 Generalized Fibonacci numbers.
• Sequence: 1, 7, 50, 357, 2549, 18200, 129949,
In OEIS: - A054413 a(n)=7*a(n-1)+a(n-2).
• Sequence: 2, 7, 51, 364, 2599, 18557, 132498,
In OEIS: - A086902 a(n) = 7a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 7, a(n) = [(7+sqrt(53))/2]^n + [(7-sqrt(53))/2]^n.
• a(n) = 8a(n-1) + a(n-2).
• Sequence: 1, 1, 9, 73, 593, 4817, 39129, 317849,
In OEIS: - A015454 Generalized Fibonacci numbers.
• Sequence: 1, 4, 33, 268, 2177, 17684, 143649,
In OEIS: - A041024 Numerators of continued fraction convergents to sqrt(17). Also A088317 a(n) = 8a(n-1) + a(n-2), starting with a(0) = 1 and a(1) = 4.
• Sequence: 1, 8, 65, 528, 4289, 34840, 283009,
In OEIS: - A041025 Denominators of continued fraction convergents to sqrt(17).
• Sequence: 2, 8, 66, 536, 4354, 35368, 287298,
In OEIS: - A086594 a(n)=8a(n-1)+a(n-2), starting with a(0)=2 and a(1)=8.
• a(n) = 9a(n-1) + a(n-2).
• Sequence: 1, 1, 10, 91, 829, 7552, 68797, 626725,
In OEIS: - A015455 Generalized Fibonacci numbers.
• Sequence: 0, 1, 9, 82, 747, 6805, 61992, 564733,
In OEIS: - A099371 Generalized Fibonacci sequence.
• Sequence: 2, 9, 83, 756, 6887, 62739, 571538,
In OEIS: - A087798 a(n) = 9a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 9, a(n) = [(9+sqrt(85))/2]^n + [(9-sqrt(85))/2]^n.
• a(n) = 10a(n-1) + a(n-2).
• Sequence: 1, 11, 111, 1121, 11321, 114331, 1154631,
In OEIS: - A015456 Generalized Fibonacci numbers.
• Sequence: 1, 4, 41, 414, 4181, 42224, 426421,
In OEIS: - A109109 a(n)=10a(n-1)+a(n-2), a(0)=1,a(1)=4.
• Sequence: 1, 5, 51, 515, 5201, 52525, 530451,
In OEIS: - A088320 a(n) = 10a(n-1) + a(n-2), starting with a(0) = 1 and a(1) = 5. Also A041040 Numerators of continued fraction convergents to sqrt(26).
• Sequence: 1, 9, 91, 919, 9281, 93729, 946571,
In OEIS: - A109108 a(n)=10a(n-1)+a(n-2), a(0)=1,a(1)=9.
• Sequence: 1, 10, 101, 1020, 10301, 104030,
In OEIS: - A041041 Denominators of continued fraction convergents to sqrt(26).
• Sequence: 2, 10, 102, 1030, 10402, 105050,
In OEIS: - A086927 a(n) = 10a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 10, a(n) = (5+sqrt(26))^n + (5-sqrt(26))^n.
• Sequence: 2, 20, 202, 2040, 20602, 208060,
In OEIS: - A109107 (1/sqrt(26))((5+sqrt(26))^(n+1)-(5-sqrt(26))^(n+1)).
• a(n) = 11a(n-1) + a(n-2).
• Sequence: 1, 1, 12, 133, 1475, 16358,
In OEIS: - A015457 Generalized Fibonacci numbers.
• Sequence: 1, 8, 89, 987, 10946, 121393,
In OEIS: - A099100 Fib(5n+1).
• Sequence: 1, 11, 122, 1353, 15005, 166408,
In OEIS: - A049666 F(5n)/5, where F=A000045 (the Fibonacci sequence).
• Sequence: 2, 11, 123, 1364, 15127, 167761,
In OEIS: - A001946 a(n) = 11a(n-1) + a(n-2).
• Sequence: 3, 29, 322, 3571, 39603, 439204,
In OEIS: - A001947 Related to Bernoulli numbers.
• Sequence: 5, 55, 610, 6765, 75025, 832040,
In OEIS: - A102312 Fibonacci(5n).
• a(n) = 12a(n-1) + a(n-2).
• Sequence: 1, 6, 73, 882, 10657, 128766, 1555849,
In OEIS: - A041060 Numerators of continued fraction convergents to sqrt(37). Also A089926 a(n)=12a(n-1)+a(n-2), a(0)=1,a(1)=6.
• Sequence: 1, 12, 145, 1752, 21169, 255780,
In OEIS: - A041061 Denominators of continued fraction convergents to sqrt(37).
• Sequence: 2, 12, 146, 1764, 21314, 257532,
In OEIS: - A086928 a(n) = 12a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 12, a(n) = (6+sqrt(37))^n + (6-sqrt(37))^n.
• a(n) = 13a(n-1) + a(n-2).
• Sequence: 2, 13, 171, 2236, 29239, 382343,
In OEIS: - A088316 a(n) = 13a(n-1) + a(n-2).
• a(n) = 14a(n-1) + a(n-2).
• Sequence: 1, 7, 99, 1393, 19601, 275807,
In OEIS: - A041084 Numerators of continued fraction convergents to sqrt(50).
• Sequence: 1, 14, 197, 2772, 39005, 548842,
In OEIS: - A041085 Denominators of continued fraction convergents to sqrt(50).
• Sequence: 2, 14, 198, 2786, 39202, 551614,
In OEIS: - A090300 a(n) = 14a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 14.
• a(n) = 15a(n-1) + a(n-2).
• Sequence: 2, 15, 227, 3420, 51527, 776325,
In OEIS: - A090301 a(n) = 15a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 15.
• a(n) = 16a(n-1) + a(n-2).
• Sequence: 1, 8, 129, 2072, 33281, 534568, 8586369,
In OEIS: - A041112 Numerators of continued fraction convergents to sqrt(65).
• Sequence: 1, 16, 257, 4128, 66305, 1065008,
In OEIS: - A041113 Denominators of continued fraction convergents to sqrt(65).
• Sequence: 2, 16, 258, 4144, 66562, 1069136,
In OEIS: - A090305 a(n) = 16a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 16.
• a(n) = 17a(n-1) + a(n-2).
• Sequence: 2, 17, 291, 4964, 84679, 1444507,
In OEIS: - A090306 a(n) = 17a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 17.
• a(n) = 18a(n-1) + a(n-2).
• Sequence: 1, 9, 163, 2943, 53137, 959409, 17322499,
In OEIS: - A041144 Numerators of continued fraction convergents to sqrt(82).
• Sequence: 1, 18, 325, 5868, 105949, 1912950, 34539049,
In OEIS: - A041145 Denominators of continued fraction convergents to sqrt(82).
• Sequence: 2, 18, 326, 5886, 106274, 1918818,
In OEIS: - A090307 a(n) = 18a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 18.
• a(n) = 19a(n-1) + a(n-2).
• Sequence: 2, 19, 363, 6916, 131767, 2510489,
In OEIS: - A090308 a(n) = 19a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 19.
• a(n) = 20a(n-1) + a(n-2).
• Sequence: 1, 10, 201, 4030, 80801, 1620050, 32481801,
In OEIS: - A041180 Numerators of continued fraction convergents to sqrt(101).
• Sequence: 1, 20, 401, 8040, 161201, 3232060, 64802401,
In OEIS: - A041181 Denominators of continued fraction convergents to sqrt(101).
• Sequence: 2, 20, 402, 8060, 161602, 3240100,
In OEIS: - A090309 a(n) = 20a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 20.
• a(n) = 21a(n-1) + a(n-2).
• Sequence: 2, 21, 443, 9324, 196247, 4130511,
In OEIS: - A090310 a(n) = 21a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 21.
• a(n) = 22a(n-1) + a(n-2).
• Sequence: 1, 11, 243, 5357, 118097, 2603491, 57394899,
In OEIS: - A041220 Numerators of continued fraction convergents to sqrt(122).
• Sequence: 1, 22, 485, 10692, 235709, 5196290, 114554089,
In OEIS: - A041221 Denominators of continued fraction convergents to sqrt(122).
• Sequence: 2, 22, 486, 10714, 236194, 5206982,
In OEIS: - A090313 a(n) = 22a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 22.
• a(n) = 23a(n-1) + a(n-2).
• Sequence: 2, 23, 531, 12236, 281959, 6497293,
In OEIS: - A090314 a(n) = 23a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 23.
• a(n) = 24a(n-1) + a(n-2).
• Sequence: 1, 12, 289, 6948, 167041, 4015932,
In OEIS: - A041264 Numerators of continued fraction convergents to sqrt(145).
• Sequence: 1, 24, 577, 13872, 333505, 8017992,
In OEIS: - A041265 Denominators of continued fraction convergents to sqrt(145).
• Sequence: 1, 27, 649, 15603, 375121, 9018507,
In OEIS: - A095898 The (1,1)-term of the 3 X 3 matrix M^n, where M=[1,2,3; 4,7,11; 6,10,16].
• Sequence: 2, 24, 578, 13896, 334082, 8031864,
In OEIS: - A090316 a(n) = 24a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 24.
• a(n) = 26a(n-1) + a(n-2).
• Sequence: 1, 13, 339, 8827, 229841, 5984693, 155831859,
In OEIS: - A041312 Numerators of continued fraction convergents to sqrt(170).
• Sequence: 1, 26, 677, 17628, 459005, 11951758,
In OEIS: - A041313 Denominators of continued fraction convergents to sqrt(170).
• a(n) = 28a(n-1) + a(n-2).
• Sequence: 1, 14, 393, 11018, 308897, 8660134,
In OEIS: - A041364 Numerators of continued fraction convergents to sqrt(197).
• Sequence: 1, 28, 785, 22008, 617009, 17298260,
In OEIS: - A041365 Denominators of continued fraction convergents to sqrt(197).
• a(n) = 29a(n-1) + a(n-2).
• Sequence: 0, 1, 29, 842, 24447, 709805, 20608792,
In OEIS: - A049667 a(n)=F(7n)/13, where F=A000045 (the Fibonacci sequence).
• Sequence: 2, 29, 843, 24476, 710647, 20633239,
In OEIS: - A087281 Lucas numbers L(7n).
• a(n) = 30a(n-1) + a(n-2).
• Sequence: 1, 15, 451, 13545, 406801, 12217575, 366934051,
In OEIS: - A041420 Numerators of continued fraction convergents to sqrt(226).
• Sequence: 1, 30, 901, 27060, 812701, 24408090, 733055401,
In OEIS: - A041421 Denominators of continued fraction convergents to sqrt(226).
• a(n) = 32a(n-1) + a(n-2).
• Sequence: 1, 16, 513, 16432, 526337, 16859216, 540021249,
In OEIS: - A041480 Numerators of continued fraction convergents to sqrt(257).
• Sequence: 1, 32, 1025, 32832, 1051649, 33685600,
In OEIS: - A041481 Denominators of continued fraction convergents to sqrt(257).
• a(n) = 34a(n-1) + a(n-2).
• Sequence: 1, 17, 579, 19703, 670481, 22816057, 776416419,
In OEIS: - A041544 Numerators of continued fraction convergents to sqrt(290).
• Sequence: 1, 34, 1157, 39372, 1339805, 45592742,
In OEIS: - A041545 Denominators of continued fraction convergents to sqrt(290).
• a(n) = 36a(n-1) + a(n-2).
• Sequence: 1, 18, 649, 23382, 842401, 30349818,
In OEIS: - A041612 Numerators of continued fraction convergents to sqrt(325).
• Sequence: 1, 36, 1297, 46728, 1683505, 60652908,
In OEIS: - A041613 Denominators of continued fraction convergents to sqrt(325).
• a(n) = 38a(n-1) + a(n-2).
• Sequence: 1, 19, 723, 27493, 1045457, 39754859,
In OEIS: - A041684 Numerators of continued fraction convergents to sqrt(362).
• Sequence: 1, 38, 1445, 54948, 2089469, 79454770,
In OEIS: - A041685 Denominators of continued fraction convergents to sqrt(362).
• a(n) = 40a(n-1) + a(n-2).
• Sequence: 1, 20, 801, 32060, 1283201, 51360100,
In OEIS: - A041760 Numerators of continued fraction convergents to sqrt(401).
• Sequence: 1, 40, 1601, 64080, 2564801, 102656120,
In OEIS: - A041761 Denominators of continued fraction convergents to sqrt(401).
• a(n) = 42a(n-1) + a(n-2).
• Sequence: 1, 21, 883, 37107, 1559377, 65530941,
In OEIS: - A041840 Numerators of continued fraction convergents to sqrt(442).
• Sequence: 1, 42, 1765, 74172, 3116989, 130987710,
In OEIS: - A041841 Denominators of continued fraction convergents to sqrt(442).
• a(n) = 44a(n-1) + a(n-2).
• Sequence: 1, 22, 969, 42658, 1877921, 82671182,
In OEIS: - A041924 Numerators of continued fraction convergents to sqrt(485).
• Sequence: 1, 44, 1937, 85272, 3753905, 165257092,
In OEIS: - A041925 Denominators of continued fraction convergents to sqrt(485).
• a(n) = 46a(n-1) + a(n-2).
• Sequence: 1, 23, 1059, 48737, 2242961, 103224943,
In OEIS: - A042012 Numerators of continued fraction convergents to sqrt(530).
• Sequence: 1, 46, 2117, 97428, 4483805, 206352458,
In OEIS: - A042013 Denominators of continued fraction convergents to sqrt(530).
• a(n) = 48a(n-1) + a(n-2).
• Sequence: 1, 24, 1153, 55368, 2658817, 127678584,
In OEIS: - A042104 Numerators of continued fraction convergents to sqrt(577).
• Sequence: 1, 48, 2305, 110688, 5315329, 255246480,
In OEIS: - A042105 Denominators of continued fraction convergents to sqrt(577).
• a(n) = 50a(n-1) + a(n-2).
• Sequence: 1, 25, 1251, 62575, 3130001, 156562625,
In OEIS: - A042200 Numerators of continued fraction convergents to sqrt(626).
• Sequence: 1, 50, 2501, 125100, 6257501, 313000150,
In OEIS: - A042201 Denominators of continued fraction convergents to sqrt(626).
• a(n) = 52a(n-1) + a(n-2).
• Sequence: 1, 26, 1353, 70382, 3661217, 190453666,
In OEIS: - A042300 Numerators of continued fraction convergents to sqrt(677).
• Sequence: 1, 52, 2705, 140712, 7319729, 380766620,
In OEIS: - A042301 Denominators of continued fraction convergents to sqrt(677).
• a(n) = 54a(n-1) + a(n-2).
• Sequence: 1, 27, 1459, 78813, 4257361, 229976307,
In OEIS: - A042404 Numerators of continued fraction convergents to sqrt(730).
• Sequence: 1, 54, 2917, 157572, 8511805, 459795042,
In OEIS: - A042405 Denominators of continued fraction convergents to sqrt(730).
• a(n) = 56a(n-1) + a(n-2).
• Sequence: 1, 28, 1569, 87892, 4923521, 275805068,
In OEIS: - A042512 Numerators of continued fraction convergents to sqrt(785).
• Sequence: 1, 56, 3137, 175728, 9843905, 551434408,
In OEIS: - A042513 Denominators of continued fraction convergents to sqrt(785).
• a(n) = 58a(n-1) + a(n-2).
• Sequence: 1, 29, 1683, 97643, 5664977, 328666309,
In OEIS: - A042624 Numerators of continued fraction convergents to sqrt(842).
• Sequence: 1, 58, 3365, 195228, 11326589, 657137390,
In OEIS: - A042625 Denominators of continued fraction convergents to sqrt(842).
• a(n) = 60a(n-1) + a(n-2).
• Sequence: 1, 30, 1801, 108090, 6487201, 389340150,
In OEIS: - A042740 Numerators of continued fraction convergents to sqrt(901).
• Sequence: 1, 60, 3601, 216120, 12970801, 778464180,
In OEIS: - A042741 Denominators of continued fraction convergents to sqrt(901).
• a(n) = 62a(n-1) + a(n-2).
• Sequence: 1, 31, 1923, 119257, 7395857, 458662391,
In OEIS: - A042860 Numerators of continued fraction convergents to sqrt(962).
• Sequence: 1, 62, 3845, 238452, 14787869, 917086330,
In OEIS: - A042861 Denominators of continued fraction convergents to sqrt(962).
• a(n) = 76a(n-1) + a(n-2).
• Sequence: 2, 76, 5778, 439204, 33385282,
In OEIS: - A087287 Lucas numbers L(9n).
• Sequence: 1, 76, 5777, 439128, 33379505,
In OEIS: - A049669 a(n)=F(9n)/34, where F=A000045 (the Fibonacci sequence).
• a(n) = 100a(n-1) + a(n-2).
• Sequence: 1, 100, 10001, 1000200, 100030001,
In OEIS: - A096885 Related to diagonals of Pascal's triangle.
• a(n) = 137a(n-1) + a(n-2).
• Sequence: 2, 137, 18771, 2571764, 352350439,
In OEIS: - A087619 a(n) = 137a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 137,.
• a(n) = 199a(n-1) + a(n-2).
• Sequence: 2, 199, 39603, 7881196, 1568397607,
In OEIS: - A089772 Lucas numbers L(11n).

### a(n) = d * a(n-1). Geometric progressions.

Common properties:

• If a(1) = d, then this sequence is the number of words of length n on alphabet {1,2, ..., d}.
• If a(1) = d, then the generating function is 1/(1-dx).
• If a(1) = d+1, then this sequence is the number of words of length n on alphabet {1,2, ..., d,d+1} avoiding consecutive repeated letters.

Sequences:

• a(n) = -2a(n-1).
• Sequence starting with 1: 1, -2, 4, -8, 16, -32, 64, ..., In OEIS: - A122803 Powers of -2.
• a(n) = -a(n-1).
• Sequence starting with 1: 1, -1, 1, -1, 1, -1, 1, ..., In OEIS: - A033999 (-1)^n.
• a(n) = a(n-1). For d = 1 see constants.
• a(n) = 2a(n-1). Autocopy sequences: their first differences are the sequence itself.
• Sequence starting with 1: 1, 2, 4, 8, 16, 32, 64, ..., In OEIS: - A000079 Powers of 2: a(n) = 2^n.
• Sequence starting with 3: 3, 6, 12, 24, 48, 96, 192, ..., In OEIS: - A007283 3*2^n. Also A081808 Numbers n such that the largest prime power in n factorization equals phi(n).
• Sequence starting with 4: 4, 8, 16, 32, 64, 128, 256, ..., In OEIS: - A020707 Pisot sequences E(4,8), L(4,8), P(4,8), T(4,8).
• Sequence starting with 5: 5, 10, 20, 40, 80, 160, 320, ..., In OEIS: - A020714 5*2^n.
• Sequence starting with 6: 6, 12, 24, 48, 96, 192, 384, ..., In OEIS: - A091629 Product of digits associated with A091628.
• Sequence starting with 7: 7, 14, 28, 56, 112, 224, 448, ..., In OEIS: - A005009 7*2^n.
• Sequence starting with 9: 7, 9, 18, 36, 72, 144, 288, 576, ..., In OEIS: - A005010 9*2^n.
• Sequence starting with 11: 11, 22, 44, 88, 176, 352, 704, ..., In OEIS: - A005015 11*2^n.
• Sequence starting with 13: 13, 26, 52, 104, 208, 416, 832, ..., In OEIS: - A005029 13*2^n.
• Sequence starting with 15: 15, 30, 60, 120, 240, 480, 960, ..., In OEIS: - A110286 a(n)=2*a(n-1), beginning with 15.
• Sequence starting with 17: 17, 34, 68, 136, 272, 544, ..., In OEIS: - A110287 a(n)=2*a(n-1), beginning with 17.
• Sequence starting with 19: 19, 38, 76, 152, 304, 608, ..., In OEIS: - A110288 a(n)=2*a(n-1), beginning with 19.
• a(n) = 3a(n-1).
• Sequence starting with 1: 1, 3, 9, 27, 81, 243, 729, ..., In OEIS: - A000244 Powers of 3.
• Sequence starting with 2: 2, 6, 18, 54, 162, 486, 1458, ..., In OEIS: - A008776 Pisot sequences E(2,6), L(2,6), P(2,6), T(2,6).
• Sequence starting with 5: 5, 15, 45, 135, 405, 1215, ..., In OEIS: - A005030 5*3^n.
• Sequence starting with 7: 7, 21, 63, 189, 567, 1701, ..., In OEIS: - A005032 7*3^n.
• Sequence starting with 8: 8, 24, 72, 216, 648, 1944, ..., In OEIS: - A005051 8*3^n.
• Sequence starting with 10: 10, 30, 90, 270, 810, 2430, ..., In OEIS: - A005052 10*3^n.
• Sequence starting with 11: 11, 33, 99, 297, 891, 2673, ..., In OEIS: - A120354 Square root of A120353.
• Sequence starting with 20: 20, 60, 180, 540, 1620, 4860, ..., In OEIS: - A116530 Each term is three times the previous term.
• a(n) = 4a(n-1).
• Sequence starting with 1: 1, 4, 16, 64, 256, 1024, 4096, ..., In OEIS: - A000302 Powers of 4.
• Sequence starting with 2: 2, 8, 32, 128, 512, 2048, 8192, ..., In OEIS: - A004171 2^(2n+1).
• Sequence starting with 6: 6, 24, 96, 384, 1536, 6144, 24576, ..., In OEIS: - A002023 6*4^n.
• Sequence starting with 7: 7, 28, 112, 448, 1792, 7168, ..., In OEIS: - A002042 7*4^n.
• Sequence starting with 9: 9, 36, 144, 576, 2304, 9216, ..., In OEIS: - A002063 9*4^n.
• Sequence starting with 10: 10, 40, 160, 640, 2560, 10240, ..., In OEIS: - A002066 10*4^n.
• Sequence starting with 11: 11, 44, 176, 704, 2816, 11264, ..., In OEIS: - A002089 11*4^n.
• a(n) = 5a(n-1).
• Sequence starting with 1: 1, 5, 25, 125, 625, 3125, ..., In OEIS: - A000351 Powers of 5.
• Sequence starting with 2: 2, 10, 50, 250, 1250, 6250, ..., In OEIS: - A020729 Pisot sequences E(2,10), L(2,10), P(2,10), T(2,10).
• Sequence starting with 7: 7, 35, 175, 875, 4375, 21875, ..., In OEIS: - A005055 7*5^n.
• a(n) = 6a(n-1).
Sequence: 1, 6, 36, 216, 1296, 7776, ..., In OEIS: - A000400 Powers of 6.
• a(n) = 7a(n-1).
• Sequence starting with 1: 1, 7, 49, 343, 2401, 16807, ..., In OEIS: - A000420 Powers of 7.
• Sequence starting with 2: 2, 14, 98, 686, 4802, 33614, ..., In OEIS: - A109808 Value of Tutte dichromatic polynomial T_G(0,1) where G is the Cartesian product of the paths P_2 and P_n (n>1).
• a(n) = 8a(n-1).
• Sequence starting with 1: 1, 8, 64, 512, 4096, 32768, ..., In OEIS: - A001018 Powers of 8.
• Sequence starting with 2: 2, 16, 128, 1024, 8192, 65536, ..., In OEIS: - A013730 2^(3n+1).
• Sequence starting with 4: 4, 32, 256, 2048, 16384, 131072, ..., In OEIS: - A013731 2^(3n+2).
• a(n) = 9a(n-1).
• Sequence starting with 1: 1, 9, 81, 729, 6561, 59049, ..., In OEIS: - A001019 Powers of 9. Also A100062 Denominator of the probability that an integer n occurs in the cumulative sums of the decimal digits of a random real number between 0 and 1.
• Sequence starting with 3: 3, 27, 243, 2187, 19683, 177147, ..., In OEIS: - A013708 3^(2n+1).
• a(n) = 10a(n-1).
• Sequence starting with 1: 1, 10, 100, 1000, 10000, 100000, ..., In OEIS: - A011557 Powers of 10.
• a(n) = 11a(n-1).
• Sequence starting with 1: 1, 11, 121, 1331, 14641, 161051, ..., In OEIS: - A001020 Powers of 11.
• a(n) = 12a(n-1).
• Sequence starting with 1: 1, 12, 144, 1728, 20736, 248832, ..., In OEIS: - A001021 Powers of 12.
• a(n) = 13a(n-1).
• Sequence starting with 1: 1, 13, 169, 2197, 28561, 371293, ..., In OEIS: - A001022 Powers of 13.
• a(n) = 14a(n-1).
• Sequence starting with 1: 1, 14, 196, 2744, 38416, 537824, ..., In OEIS: - A001023 Powers of 14.
• a(n) = 15a(n-1).
• Sequence starting with 1: 1, 15, 225, 3375, 50625, 759375, ..., In OEIS: - A001024 Powers of 15.
• a(n) = 16a(n-1).
• Sequence starting with 1: 1, 16, 256, 4096, 65536, 1048576, ..., In OEIS: - A001025 Powers of 16.
• Sequence starting with 2: 2, 32, 512, 8192, 131072, 2097152, ..., In OEIS: - A013776 2^(4n+1).
• Sequence starting with 4: 4, 64, 1024, 16384, 262144, ..., In OEIS: - A013709 4^(2n+1).
• Sequence starting with 8: 8, 128, 2048, 32768, 524288, ..., In OEIS: - A013777 2^(4n+3).
• a(n) = 17a(n-1).
• Sequence starting with 1: 1, 17, 289, 4913, 83521, 1419857, ..., In OEIS: - A001026 Powers of 17.
• a(n) = 18a(n-1).
• Sequence starting with 1: 1, 18, 324, 5832, 104976, 1889568, ..., In OEIS: - A001027 Powers of 18.
• a(n) = 19a(n-1).
• Sequence starting with 1: 1, 19, 361, 6859, 130321, 2476099, ..., In OEIS: - A001029 Powers of 19.
• a(n) = 20a(n-1).
• Sequence starting with 1: 1, 20, 400, 8000, 160000, 3200000, ..., In OEIS: - A009964 Powers of 20.
• a(n) = 21a(n-1).
• Sequence starting with 1: 1, 21, 441, 9261, 194481, 4084101, ..., In OEIS: - A009965 Powers of 21.
• a(n) = 22a(n-1).
• Sequence starting with 1: 1, 22, 484, 10648, 234256, 5153632, ..., In OEIS: - A009966 Powers of 22.
• a(n) = 23a(n-1).
• Sequence starting with 1: 1, 23, 529, 12167, 279841, 6436343, ..., In OEIS: - A009967 Powers of 23.
• a(n) = 24a(n-1).
• Sequence starting with 1: 1, 24, 576, 13824, 331776, 7962624, ..., In OEIS: - A009968 Powers of 24.
• a(n) = 25a(n-1).
• Sequence starting with 1: 1, 25, 625, 15625, 390625, 9765625, ..., In OEIS: - A009969 Powers of 25.
• Sequence starting with 5: 5, 125, 3125, 78125, 1953125, ..., In OEIS: - A013710 5^(2n+1).
• a(n) = 26a(n-1).
• Sequence starting with 1: 1, 26, 676, 17576, 456976, 11881376, ..., In OEIS: - A009970 Powers of 26.
• a(n) = 27a(n-1).
• Sequence starting with 1: 1, 27, 729, 19683, 531441, 14348907, ..., In OEIS: - A009971 Powers of 27.
• Sequence starting with 3: 3, 81, 2187, 59049, 1594323, ..., In OEIS: - A013732 3^(3n+1).
• Sequence starting with 9: 9, 243, 6561, 177147, 4782969, ..., In OEIS: - A013733 3^(3n+2).
• a(n) = 28a(n-1).
• Sequence starting with 1: 1, 28, 784, 21952, 614656, 17210368, ..., In OEIS: - A009972 Powers of 28.
• a(n) = 29a(n-1).
• Sequence starting with 1: 1, 29, 841, 24389, 707281, 20511149, ..., In OEIS: - A009973 Powers of 29.
• a(n) = 30a(n-1).
• Sequence starting with 1: 1, 30, 900, 27000, 810000, 24300000, ..., In OEIS: - A009974 Powers of 30.
• a(n) = 31a(n-1).
• Sequence starting with 1: 1, 31, 961, 29791, 923521, 28629151, ..., In OEIS: - A009975 Powers of 31.
• a(n) = 32a(n-1).
• Sequence starting with 1: 1, 32, 1024, 32768, 1048576, ..., In OEIS: - A009976 Powers of 32.
• Sequence starting with 2: 2, 64, 2048, 65536, 2097152, ..., In OEIS: - A013822 2^(5n+1).
• Sequence starting with 4: 4, 128, 4096, 131072, 4194304, ..., In OEIS: - A013823 2^(5n+2).
• Sequence starting with 8: 8, 256, 8192, 262144, 8388608, ..., In OEIS: - A013824 2^(5n+3).
• Sequence starting with 16: 16, 512, 16384, 524288, 16777216, ..., In OEIS: - A013825 2^(5n+4).
• a(n) = 33a(n-1).
• Sequence starting with 1: 1, 33, 1089, 35937, 1185921, ..., In OEIS: - A009977 Powers of 33.
• a(n) = 34a(n-1).
• Sequence starting with 1: 1, 34, 1156, 39304, 1336336, ..., In OEIS: - A009978 Powers of 34.
• a(n) = 35a(n-1).
• Sequence starting with 1: 1, 35, 1225, 42875, 1500625, ..., In OEIS: - A009979 Powers of 35.
• a(n) = 36a(n-1).
• Sequence starting with 1: 1, 36, 1296, 46656, 1679616, ..., In OEIS: - A009980 Powers of 36.
• Sequence starting with 6: 6, 216, 7776, 279936, 10077696, ..., In OEIS: - A013711 6^(2n+1).
• a(n) = 37a(n-1).
• Sequence starting with 1: 1, 37, 1369, 50653, 1874161, ..., In OEIS: - A009981 Powers of 37.
• a(n) = 38a(n-1).
• Sequence starting with 1: 1, 38, 1444, 54872, 2085136, ..., In OEIS: - A009982 Powers of 38.
• a(n) = 39a(n-1).
• Sequence starting with 1: 1, 39, 1521, 59319, 2313441, ..., In OEIS: - A009983 Powers of 39.
• Sequence starting with 17: 17, 663, 25857, 1008423, ..., In OEIS: - A063941 17*39^n.
• a(n) = 40a(n-1).
• Sequence starting with 1: 1, 40, 1600, 64000, 2560000, ..., In OEIS: - A009984 Powers of 40.
• a(n) = 41a(n-1).
• Sequence starting with 1: 1, 41, 1681, 68921, 2825761, ..., In OEIS: - A009985 Powers of 41.
• a(n) = 42a(n-1).
• Sequence starting with 1: 1, 42, 1764, 74088, 3111696, ..., In OEIS: - A009986 Powers of 42.
• a(n) = 43a(n-1).
• Sequence starting with 1: 1, 43, 1849, 79507, 3418801, ..., In OEIS: - A009987 Powers of 43.
• a(n) = 44a(n-1).
• Sequence starting with 1: 1, 44, 1936, 85184, 3748096, ..., In OEIS: - A009988 Powers of 44.
• a(n) = 45a(n-1).
• Sequence starting with 1: 1, 45, 2025, 91125, 4100625, ..., In OEIS: - A009989 Powers of 45.
• a(n) = 46a(n-1).
• Sequence starting with 1: 1, 46, 2116, 97336, 4477456, ..., In OEIS: - A009990 Powers of 46.
• a(n) = 47a(n-1).
• Sequence starting with 1: 1, 47, 2209, 103823, 4879681, ..., In OEIS: - A009991 Powers of 47.
• a(n) = 48a(n-1).
• Sequence starting with 1: 1, 48, 2304, 110592, 5308416, ..., In OEIS: - A009992 Powers of 48.
• a(n) = 49a(n-1).
• Sequence starting with 1: 1, 49, 2401, 117649, 5764801, ..., In OEIS: - A087752 Powers of 49.
• Sequence starting with 7: 7, 343, 16807, 823543, 40353607, ..., In OEIS: - A013712 7^(2n+1).
• a(n) = 64a(n-1).
• Sequence starting with 1: 1, 64, 4096, 262144, 16777216, ..., In OEIS: - A089357 2^(6n).
• Sequence starting with 4: 4, 256, 16384, 1048576, 67108864, ..., In OEIS: - A013734 4^(3n+1).
• Sequence starting with 8: 8, 512, 32768, 2097152, 134217728, ..., In OEIS: - A013713 8^(2n+1).
• Sequence starting with 16: 16, 1024, 65536, 4194304, 268435456, ..., In OEIS: - A013735 4^(3n+2).
• a(n) = 81a(n-1).
• Sequence starting with 3: 3, 243, 19683, 1594323, 129140163, ..., In OEIS: - A013778 3^(4n+1).
• Sequence starting with 9: 9, 729, 59049, 4782969, 387420489, ..., In OEIS: - A013714 9^(2n+1).
• Sequence starting with 27: 27, 2187, 177147, 14348907, ..., In OEIS: - A013779 3^(4n+3).
• Sequence starting with 81: 81, 6561, 531441, 43046721, ..., In OEIS: - A089683 3^(4n).
• a(n) = 100a(n-1).
• Sequence starting with 1: 1, 100, 10000, 1000000, ..., In OEIS: - A098608 100^n.
• Sequence starting with 10: 10, 1000, 100000, 10000000, ..., In OEIS: - A013715 10^(2n+1).
• a(n) = 101a(n-1).
• Sequence starting with 1: 1, 101, 10201, 1030301, 104060401, ..., In OEIS: - A096884 Related to Pascal's triangle.
• a(n) = 121a(n-1).
• Sequence starting with 11: 11, 1331, 161051, 19487171, ..., In OEIS: - A013716 11^(2n+1).
• a(n) = 125a(n-1).
• Sequence starting with 5: 5, 625, 78125, 9765625, 1220703125, ..., In OEIS: - A013736 5^(3n+1).
• Sequence starting with 25: 25, 3125, 390625, 48828125, 6103515625, ..., In OEIS: - A013737 5^(3n+2).
• a(n) = 144a(n-1).
• Sequence starting with 12: 12, 1728, 248832, 35831808, 5159780352, ..., In OEIS: - A013717 12^(2n+1).
• a(n) = 169a(n-1).
• Sequence starting with 13: 13, 2197, 371293, 62748517, 10604499373, ..., In OEIS: - A013718 13^(2n+1).
• a(n) = 196a(n-1).
• Sequence starting with 14: 14, 2744, 537824, 105413504, 20661046784, ..., In OEIS: - A013719 14^(2n+1).
• a(n) = 216a(n-1).
• Sequence starting with 6: 6, 1296, 279936, 60466176, 13060694016, ..., In OEIS: - A013738 6^(3n+1).
• Sequence starting with 36: 36, 7776, 1679616, 362797056, ..., In OEIS: - A013739 6^(3n+2).
• a(n) = 225a(n-1).
• Sequence starting with 15: 15, 3375, 759375, 170859375, 38443359375, ..., In OEIS: - A013720 15^(2n+1).
• a(n) = 243a(n-1).
• Sequence starting with 3: 3, 729, 177147, 43046721, ..., In OEIS: - A013826 3^(5n+1).
• Sequence starting with 9: 9, 2187, 531441, 129140163, ..., In OEIS: - A013827 3^(5n+2).
• Sequence starting with 81: 9, 81, 19683, 4782969, 1162261467, ..., In OEIS: - A013829 3^(5n+4).
• a(n) = 256a(n-1).
• Sequence starting with 4: 4, 1024, 262144, 67108864, ..., In OEIS: - A013780 4^(4n+1).
• Sequence starting with 16: 16, 4096, 1048576, 268435456, ..., In OEIS: - A013721 16^(2n+1).
• Sequence starting with 64: 64, 16384, 4194304, 1073741824, ..., In OEIS: - A013781 4^(4n+3).
• a(n) = 289a(n-1).
• Sequence starting with 17: 17, 4913, 1419857, 410338673, ..., In OEIS: - A013722 17^(2n+1).
• a(n) = 324a(n-1).
• Sequence starting with 18: 18, 5832, 1889568, 612220032, ..., In OEIS: - A013723 18^(2n+1).
• a(n) = 343a(n-1).
• Sequence starting with 7: 7, 2401, 823543, 282475249, ..., In OEIS: - A013740 7^(3n+1).
• Sequence starting with 49: 49, 16807, 5764801, 1977326743, ..., In OEIS: - A013741 7^(3n+2).
• a(n) = 361a(n-1).
• Sequence starting with 19: 19, 6859, 2476099, 893871739, ..., In OEIS: - A013724 19^(2n+1).
• a(n) = 400a(n-1).
• Sequence starting with 20: 120, 8000, 3200000, 1280000000, ..., In OEIS: - A013725 20^(2n+1).
• a(n) = 441a(n-1).
• Sequence starting with 21: 21, 9261, 4084101, 1801088541, ..., In OEIS: - A013726 21^(2n+1).
• a(n) = 484a(n-1).
• Sequence starting with 22: 22, 10648, 5153632, 2494357888, ..., In OEIS: - A013727 22^(2n+1).
• a(n) = 512a(n-1).
• Sequence starting with 8: 8, 4096, 2097152, 1073741824, ..., In OEIS: - A013742 8^(3n+1).
• Sequence starting with 64: 64, 32768, 16777216, 8589934592, ..., In OEIS: - A013743 8^(3n+2).
• a(n) = 529a(n-1).
• Sequence starting with 23: 23, 12167, 6436343, 3404825447, ..., In OEIS: - A013728 23^(2n+1).
• a(n) = 576a(n-1).
• Sequence starting with 24: 24, 13824, 7962624, 4586471424, ..., In OEIS: - A013729 24^(2n+1).
• a(n) = 625a(n-1).
• Sequence starting with 5: 5, 3125, 1953125, 1220703125, ..., In OEIS: - A013782 5^(4n+1).
• Sequence starting with 125: 125, 78125, 48828125, ..., In OEIS: - A013783 5^(4n+3).
• a(n) = 729a(n-1).
• Sequence starting with 9: 9, 6561, 4782969, 3486784401, ..., In OEIS: - A013744 9^(3n+1).
• Sequence starting with 27: 27, 6561, 1594323, 387420489, ..., In OEIS: - A013828 3^(5n+3).
• Sequence starting with 81: 81, 59049, 43046721, 31381059609, ..., In OEIS: - A013745 9^(3n+2).
• a(n) = 1000a(n-1).
• Sequence starting with 10: 10, 10000, 10000000, 10000000000, ..., In OEIS: - A013746 10^(3n+1).
• Sequence starting with 100: 100, 100000, 100000000, ..., In OEIS: - A013747 10^(3n+2).
• a(n) = 1001a(n-1).
• Sequence starting with 1: 1, 1001, 1002001, 1003003001, 1004006004001, ..., In OEIS: - A097659 1001^n.
• a(n) = 1024a(n-1).
• Sequence starting with 4: 4, 4096, 4194304, 4294967296, ..., In OEIS: - A013830 4^(5n+1).
• Sequence starting with 16: 16, 16384, 16777216, 17179869184, ..., In OEIS: - A013831 4^(5n+2).
• Sequence starting with 64: 64, 65536, 67108864, 68719476736, ..., In OEIS: - A013832 4^(5n+3).
• Sequence starting with 256: 256, 262144, 268435456, 274877906944, ..., In OEIS: - A013833 4^(5n+4).
• a(n) = 1296a(n-1).
• Sequence starting with 6: 6, 7776, 10077696, 13060694016, ..., In OEIS: - A013784 6^(4n+1).
• Sequence starting with 216: 216, 279936, 362797056, 470184984576, ..., In OEIS: - A013785 6^(4n+3).
• a(n) = 1331a(n-1).
• Sequence starting with 11: 11, 14641, 19487171, 25937424601, ..., In OEIS: - A013748 11^(3n+1).
• Sequence starting with 121: 121, 161051, 214358881, ..., In OEIS: - A013749 11^(3n+2).
• a(n) = 1728a(n-1).
• Sequence starting with 12: 12, 20736, 35831808, 61917364224, ..., In OEIS: - A013750 12^(3n+1).
• Sequence starting with 144: 144, 248832, 429981696, ..., In OEIS: - A013751 12^(3n+2).
• a(n) = 2197a(n-1).
• Sequence starting with 13: 13, 28561, 62748517, 137858491849, ..., In OEIS: - A013752 13^(3n+1).
• Sequence starting with 169: 169, 371293, 815730721, ..., In OEIS: - A013753 13^(3n+2).
• a(n) = 2401a(n-1).
• Sequence starting with 7: 7, 16807, 40353607, 96889010407, ..., In OEIS: - A013786 7^(4n+1).
• Sequence starting with 343: 343, 823543, 1977326743, 4747561509943,..., In OEIS: - A013787 7^(4n+3).
• a(n) = 2744a(n-1).
• Sequence starting with 14: 14, 38416, 105413504, 289254654976, ..., In OEIS: - A013754 14^(3n+1).
• Sequence starting with 196: 196, 537824, 1475789056, ..., In OEIS: - A013755 14^(3n+2).
• a(n) = 3125a(n-1).
• Sequence starting with 5: 5, 15625, 48828125, 152587890625, ..., In OEIS: - A013834 5^(5n+1).
• Sequence starting with 25: 25, 78125, 244140625, 762939453125, ..., In OEIS: - A013835 5^(5n+2).
• Sequence starting with 125: 125, 390625, 1220703125, ..., In OEIS: - A013836 5^(5n+3).
• Sequence starting with 625: 625, 1953125, 6103515625, ..., In OEIS: - A013837 5^(5n+4).
• a(n) = 3375a(n-1).
• Sequence starting with 15: 15, 50625, 170859375, 576650390625, ..., In OEIS: - A013756 15^(3n+1).
• Sequence starting with 225: 225, 759375, 2562890625, 8649755859375, ..., In OEIS: - A013757 15^(3n+2).
• a(n) = 4096a(n-1).
• Sequence starting with 8: 8, 32768, 134217728, 549755813888, ..., In OEIS: - A013788 8^(4n+1).
• Sequence starting with 16: 16, 65536, 268435456, 1099511627776, ..., In OEIS: - A013758 16^(3n+1).
• Sequence starting with 256: 256, 1048576, 4294967296, 17592186044416, ..., In OEIS: - A013759 16^(3n+2).
• Sequence starting with 512: 512, 2097152, 8589934592, 35184372088832, ..., In OEIS: - A013789 8^(4n+3).
• a(n) = 4913a(n-1).
• Sequence starting with 17: 17, 83521, 410338673, 2015993900449, ..., In OEIS: - A013760 17^(3n+1).
• Sequence starting with 289: 289, 1419857, 6975757441, 34271896307633, ..., In OEIS: - A013761 17^(3n+2).
• a(n) = 5832a(n-1).
• Sequence starting with 18: 18, 104976, 612220032, 3570467226624, ..., In OEIS: - A013762 18^(3n+1).
• Sequence starting with 324: 324, 1889568, 11019960576, 64268410079232, ..., In OEIS: - A013763 18^(3n+2).
• a(n) = 6561a(n-1).
• Sequence starting with 9: 9, 59049, 387420489, 2541865828329, ..., In OEIS: - A013790 9^(4n+1).
• Sequence starting with 729: 729, 4782969, 31381059609, 205891132094649, ..., In OEIS: - A013791 9^(4n+3).
• a(n) = 6859a(n-1).
• Sequence starting with 19: 19, 130321, 893871739, 6131066257801, ..., In OEIS: - A013764 19^(3n+1).
• Sequence starting with 361: 361, 2476099, 16983563041, 116490258898219, ..., In OEIS: - A013765 19^(3n+2).
• a(n) = 7776a(n-1).
• Sequence starting with 6: 6, 46656, 362797056, 2821109907456, ..., In OEIS: - A013838 6^(5n+1).
• Sequence starting with 36: 36, 279936, 2176782336, 16926659444736, ..., In OEIS: - A013839 6^(5n+2).
• Sequence starting with 216: 216, 1679616, 13060694016, 101559956668416, ..., In OEIS: - A013840 6^(5n+3).
• Sequence starting with 1296: 1296, 10077696, 78364164096, ..., In OEIS: - A013841 6^(5n+4).
• a(n) = 8000a(n-1).
• Sequence starting with 20: 20, 160000, 1280000000, 10240000000000, ..., In OEIS: - A013766 20^(3n+1).
• Sequence starting with 400: 400, 3200000, 25600000000, 204800000000000, ..., In OEIS: - A013767 20^(3n+2).
• a(n) = 9261a(n-1).
• Sequence starting with 21: 21, 194481, 1801088541, 16679880978201, ..., In OEIS: - A013768 21^(3n+1).
• Sequence starting with 441: 441, 4084101, 37822859361, 350277500542221, ..., In OEIS: - A013769 21^(3n+2).
• a(n) = 10000a(n-1).
• Sequence starting with 10: 10, 100000, 1000000000, 10000000000000, ..., In OEIS: - A013792 10^(4n+1).
• Sequence starting with 1000: 1000, 10000000, 100000000000, ..., In OEIS: - A013793 10^(4n+3).
• a(n) = 10648a(n-1).
• Sequence starting with 22: 22, 234256, 2494357888, 26559922791424, ..., In OEIS: - A013770 22^(3n+1).
• Sequence starting with 484: 484, 5153632, 54875873536, ..., In OEIS: - A013771 22^(3n+2).
• a(n) = 12167a(n-1).
• Sequence starting with 23: 23, 279841, 3404825447, 41426511213649, ..., In OEIS: - A013772 23^(3n+1).
• Sequence starting with 529: 529, 6436343, 78310985281, 952809757913927, ..., In OEIS: - A013773 23^(3n+2).
• a(n) = 13824a(n-1).
• Sequence starting with 24: 24, 331776, 4586471424, 63403380965376,..., In OEIS: - A013774 24^(3n+1).
• Sequence starting with 576: 576, 7962624, 110075314176, ..., In OEIS: - A013775 24^(3n+2).
• a(n) = 14641a(n-1).
• Sequence starting with 11: 11, 161051, 2357947691, 34522712143931, ..., In OEIS: - A013794 11^(4n+1).
• Sequence starting with 1331: 1331, 19487171, 285311670611, ..., In OEIS: - A013795 11^(4n+3).
• a(n) = 16807a(n-1).
• Sequence starting with 7: 7, 117649, 1977326743, 33232930569601, ..., In OEIS: - A013842 7^(5n+1).
• Sequence starting with 49: 49, 823543, 13841287201, 232630513987207, ..., In OEIS: - A013843 7^(5n+2).
• Sequence starting with 343: 343, 5764801, 96889010407, 1628413597910449, ..., In OEIS: - A013844 7^(5n+3).
• Sequence starting with 2401: 2401, 40353607, 678223072849, ..., In OEIS: - A013845 7^(5n+4).
• a(n) = 20736a(n-1).
• Sequence starting with 12: 12, 248832, 5159780352, 106993205379072, ..., In OEIS: - A013796 12^(4n+1).
• Sequence starting with 1728: 1728, 35831808, 743008370688, ..., In OEIS: - A013797 12^(4n+3).
• a(n) = 28561a(n-1).
• Sequence starting with 13: 13, 371293, 10604499373, 302875106592253, ..., In OEIS: - A013798 13^(4n+1).
• Sequence starting with 2197: 2197, 62748517, 1792160394037, ..., In OEIS: - A013799 13^(4n+3).
• a(n) = 32768a(n-1).
• Sequence starting with 8: 8, 262144, 8589934592, 281474976710656, ..., In OEIS: - A013846 8^(5n+1).
• Sequence starting with 64: 64, 2097152, 68719476736, ..., In OEIS: - A013847 8^(5n+2).
• Sequence starting with 512: 512, 16777216, 549755813888, ..., In OEIS: - A013848 8^(5n+3).
• Sequence starting with 4096: 4096, 134217728, 4398046511104, ..., In OEIS: - A013849 8^(5n+4).
• a(n) = 38416a(n-1).
• Sequence starting with 14: 14, 537824, 20661046784, 793714773254144, ..., In OEIS: - A013800 14^(4n+1).
• Sequence starting with 2744: 2744, 105413504, 4049565169664, ..., In OEIS: - A013801 14^(4n+3).
• a(n) = 50625a(n-1).
• Sequence starting with 15: 15, 759375, 38443359375, 1946195068359375, ..., In OEIS: - A013802 15^(4n+1).
• Sequence starting with 3375: 3375, 170859375, 8649755859375, ..., In OEIS: - A013803 15^(4n+3).
• a(n) = 59049a(n-1).
• Sequence starting with 9: 9, 531441, 31381059609, 1853020188851841, ..., In OEIS: - A013850 9^(5n+1).
• Sequence starting with 81: 81, 4782969, 282429536481, ..., In OEIS: - A013851 9^(5n+2).
• Sequence starting with 729: 8729, 43046721, 2541865828329, ..., In OEIS: - A013852 9^(5n+3).
• Sequence starting with 6561: 6561, 387420489, 22876792454961, ..., In OEIS: - A013853 9^(5n+4).
• a(n) = 65536a(n-1).
• Sequence starting with 16: 16, 1048576, 68719476736, 4503599627370496, ..., In OEIS: - A013804 16^(4n+1).
• Sequence starting with 4096: 4096, 268435456, 17592186044416, ..., In OEIS: - A013805 16^(4n+3).
• a(n) = 83521a(n-1).
• Sequence starting with 17: 17, 1419857, 118587876497, 9904578032905937, ..., In OEIS: - A013806 17^(4n+1).
• Sequence starting with 4913: 4913, 410338673, 34271896307633, ..., In OEIS: - A013807 17^(4n+3).
• a(n) = 100000a(n-1).
• Sequence starting with 10: 10, 1000000, 100000000000, 10000000000000000, ..., In OEIS: - A013854 10^(5n+1).
• Sequence starting with 100: 100, 10000000, 1000000000000, ..., In OEIS: - A013855 10^(5n+2).
• Sequence starting with 1000: 1000, 100000000, 10000000000000, ..., In OEIS: - A013856 10^(5n+3).
• Sequence starting with 10000: 10000, 1000000000, 100000000000000, ..., In OEIS: - A013857 10^(5n+4).
• a(n) = 104976a(n-1).
• Sequence starting with 18: 18, 1889568, 198359290368, 20822964865671168, ..., In OEIS: - A013808 18^(4n+1).
• Sequence starting with 5832: 5832, 612220032, 64268410079232, ..., In OEIS: - A013809 18^(4n+3).
• a(n) = 130321a(n-1).
• Sequence starting with 19: 19, 2476099, 322687697779, 42052983462257059, ..., In OEIS: - A013810 19^(4n+1).
• Sequence starting with 6859: 6859, 893871739, 116490258898219, ..., In OEIS: - A013811 19^(4n+3).
• a(n) = 160000a(n-1).
• Sequence starting with 20: 20, 3200000, 512000000000, 81920000000000000, ..., In OEIS: - A013812 20^(4n+1).
• Sequence starting with 8000: 8000, 1280000000, 204800000000000, ..., In OEIS: - A013813 20^(4n+3).
• a(n) = 161051a(n-1).
• Sequence starting with 11: 11, 1771561, 285311670611, 45949729863572161, ..., In OEIS: - A013858 11^(5n+1).
• Sequence starting with 121: 121, 19487171, 3138428376721, ..., In OEIS: - A013859 11^(5n+2).
• Sequence starting with 1331: 1331, 214358881, 34522712143931, ..., In OEIS: - A013860 11^(5n+3).
• Sequence starting with 14641: 14641, 2357947691, 379749833583241, ..., In OEIS: - A013861 11^(5n+4).
• a(n) = 194481a(n-1).
• Sequence starting with 21: 21, 4084101, 794280046581, ..., In OEIS: - A013814 21^(4n+1).
• Sequence starting with 9261: 9261, 1801088541, 350277500542221, ..., In OEIS: - A013815 21^(4n+3).
• a(n) = 234256a(n-1).
• Sequence starting with 22: 22, 5153632, 1207269217792, ..., In OEIS: - A013816 22^(4n+1).
• Sequence starting with 10648: 10648, 2494357888, 584318301411328, ..., In OEIS: - A013817 22^(4n+3).
• a(n) = 248832a(n-1).
• Sequence starting with 12: 12, 2985984, 743008370688, ..., In OEIS: - A013862 12^(5n+1).
• Sequence starting with 144: 144, 35831808, 8916100448256, ..., In OEIS: - A013863 12^(5n+2).
• Sequence starting with 1728: 1728, 429981696, 106993205379072, ..., In OEIS: - A013864 12^(5n+3).
• Sequence starting with 248832: 20736, 5159780352, 1283918464548864, ..., In OEIS: - A013865 12^(5n+4).
• a(n) = 279841a(n-1).
• Sequence starting with 23: 23, 6436343, 1801152661463, ..., In OEIS: - A013818 23^(4n+1).
• Sequence starting with 12167: 12167, 3404825447, 952809757913927, ..., In OEIS: - A013819 23^(4n+3).
• a(n) = 331776a(n-1).
• Sequence starting with 24: 24, 7962624, 2641807540224, ..., In OEIS: - A013820 24^(4n+1).
• Sequence starting with 13824: 13824, 4586471424, 1521681143169024, ..., In OEIS: - A013821 24^(4n+3).
• a(n) = 371293a(n-1).
• Sequence starting with 13: 13, 4826809, 1792160394037, ..., In OEIS: - A013866 13^(5n+1).
• Sequence starting with 169: 169, 62748517, 23298085122481, ..., In OEIS: - A013867 13^(5n+2).
• Sequence starting with 2197: 2197, 815730721, 302875106592253, ..., In OEIS: - A013868 13^(5n+3).
• Sequence starting with 28561: 28561, 10604499373, 3937376385699289, ..., In OEIS: - A013869 13^(5n+4).
• a(n) = 537824a(n-1).
• Sequence starting with 14: 14, 7529536, 4049565169664, ..., In OEIS: - A013870 14^(5n+1).
• Sequence starting with 196: 196, 105413504, 56693912375296, ..., In OEIS: - A013871 14^(5n+2).
• Sequence starting with 2744: 2744, 1475789056, 793714773254144, ..., In OEIS: - A013872 14^(5n+3).
• Sequence starting with 38416: 38416, 20661046784, 11112006825558016, ..., In OEIS: - A013873 14^(5n+4).
• a(n) = 759375a(n-1).
• Sequence starting with 15: 15, 11390625, 8649755859375, ..., In OEIS: - A013874 15^(5n+1).
• Sequence starting with 225: 225, 170859375, 129746337890625, ..., In OEIS: - A013875 15^(5n+2).
• Sequence starting with 3375: 3375, 2562890625, 1946195068359375, ..., In OEIS: - A013876 15^(5n+3).
• Sequence starting with 50625: 50625, 38443359375, 29192926025390625, ..., In OEIS: - A013877 15^(5n+4).
• a(n) = 1048576a(n-1).
• Sequence starting with 16: 16, 16777216, 17592186044416, ..., In OEIS: - A013878 16^(5n+1).
• Sequence starting with 256: 256, 268435456, 281474976710656, ..., In OEIS: - A013879 16^(5n+2).
• Sequence starting with 4096: 4096, 4294967296, 4503599627370496, ..., In OEIS: - A013880 16^(5n+3).
• Sequence starting with 65536: 65536, 68719476736, 72057594037927936, ..., In OEIS: - A013881 16^(5n+4).
• a(n) = 1419857a(n-1).
• Sequence starting with 17: 17, 24137569, 34271896307633, ..., In OEIS: - A013882 17^(5n+1).
• Sequence starting with 289: 289, 410338673, 582622237229761, ..., In OEIS: - A013883 17^(5n+2).
• Sequence starting with 4913: 4913, 6975757441, 9904578032905937, ..., In OEIS: - A013884 17^(5n+3).
• Sequence starting with 83521: 83521, 118587876497, 168377826559400929, ..., In OEIS: - A013885 17^(5n+4).
• a(n) = 1889568a(n-1).
• Sequence starting with 18: 18, 34012224, 64268410079232, ..., In OEIS: - A013886 18^(5n+1).
• Sequence starting with 324: 324, 612220032, 1156831381426176, ..., In OEIS: - A013887 18^(5n+2).
• Sequence starting with 5832: 5832, 11019960576, 20822964865671168, ..., In OEIS: - A013888 18^(5n+3).
• Sequence starting with 104976: 104976, 198359290368, 374813367582081024, ..., In OEIS: - A013889 18^(5n+4).
• a(n) = 2476099a(n-1).
• Sequence starting with 18: 19, 47045881, 116490258898219, ..., In OEIS: - A013890 19^(5n+1).
• Sequence starting with 361: 361, 893871739, 2213314919066161, ..., In OEIS: - A013891 19^(5n+2).
• Sequence starting with 6859: 6859, 16983563041, 42052983462257059, ..., In OEIS: - A013892 19^(5n+3).
• Sequence starting with 130321: 130321, 322687697779, 799006685782884121, ..., In OEIS: - A013893 19^(5n+4).
• a(n) = 3200000a(n-1).
• Sequence starting with 20: 20, 64000000, 204800000000000, ..., In OEIS: - A013894 20^(5n+1).
• Sequence starting with 400: 400, 1280000000, 4096000000000000, ..., In OEIS: - A013895 20^(5n+2).
• Sequence starting with 8000: 8000, 25600000000, 81920000000000000, ..., In OEIS: - A013896 20^(5n+3).
• Sequence starting with 16000: 160000, 512000000000, 1638400000000000000, ..., In OEIS: - A013897 20^(5n+4).
• a(n) = 4084101a(n-1).
• Sequence starting with 21: 21, 85766121, 350277500542221, ..., In OEIS: - A013898 21^(5n+1).
• Sequence starting with 441: 441, 1801088541, 7355827511386641, ..., In OEIS: - A013899 21^(5n+2).
• Sequence starting with 9261: 9261, 37822859361, 154472377739119461, ..., In OEIS: - A013900 21^(5n+3).
• Sequence starting with 194481: 194481, 794280046581, 3243919932521508681, ..., In OEIS: - A013901 21^(5n+4).
• a(n) = 5153632a(n-1).
• Sequence starting with 22: 22, 113379904, 584318301411328, ..., In OEIS: - A013902 22^(5n+1).
• Sequence starting with 484: 484, 2494357888, 12855002631049216, ..., In OEIS: - A013903 22^(5n+2).
• Sequence starting with 10648: 10648, 54875873536, 282810057883082752, ..., In OEIS: - A013904 22^(5n+3).
• Sequence starting with 234256: 234256, 1207269217792, 6221821273427820544, ..., In OEIS: - A013905 22^(5n+4).
• a(n) = 6436343a(n-1).
• Sequence starting with 23: 23, 148035889, 952809757913927, ..., In OEIS: - A013906 23^(5n+1).
• Sequence starting with 529: 529, 3404825447, 21914624432020321, ..., In OEIS: - A013907 23^(5n+2).
• Sequence starting with 12167: 12167, 78310985281, 504036361936467383, ..., In OEIS: - A013908 23^(5n+3).
• Sequence starting with 279841: 279841, 1801152661463, 11592836324538749809, ..., In OEIS: - A013909 23^(5n+4).
• a(n) = 7962624a(n-1).
• Sequence starting with 24: 24, 191102976, 1521681143169024, ..., In OEIS: - A013910 24^(5n+1).
• Sequence starting with 576: 576, 4586471424, 36520347436056576, ..., In OEIS: - A013911 24^(5n+2).
• Sequence starting with 13824: 13824, 110075314176, 876488338465357824, ..., In OEIS: - A013912 24^(5n+3).
• Sequence starting with 331776: 331776, 2641807540224, 21035720123168587776, ..., In OEIS: - A013913 24^(5n+4).

### a(n) = a(n-1) + d. Arithmetic progressions.

Sequences:

• a(n) = a(n-1) - 1.
Sequence: -1, -2, -3, -4, -5, -6, -7, ..., shifted
In OEIS:
• a(n) = a(n-1). For d = 0 see constants.
• a(n) = a(n-1) + 1.
Sequence: 1, 2, 3, 4, 5, 6, 7, ..., shifted
In OEIS:
• Starting with 9 - A020723 Pisot sequences E(9,10), P(9,10), T(9,10).
• Starting with 8 - A020722 Pisot sequences E(8,9), P(8,9), T(8,9).
• Starting with 7 - A020719 Pisot sequences E(7,8), P(7,8), T(7,8).
• Starting with 6 - A020715 Pisot sequences E(6,7), P(6,7), T(6,7).
• Starting with 5 - A020710 Pisot sequences E(5,6), P(5,6), T(5,6).
• Starting with 4 - A020705 Pisot sequences E(4,5), P(4,5), T(4,5).
• Starting with 3 - A009056 Numbers ≥ 3.
• Starting with 2 - A020725 Integers ≥ 2.
• Starting with 1 - A000027 The natural numbers.
• Starting with 0 - A001477 The nonnegative integers.
• Starting with -1 - A023443 n-1.
• Starting with -2 - A023444 n-2.
• Starting with -3 - A023445 n-3.
• Starting with -4 - A023446 n-4.
• Starting with -5 - A023447 n-5.
• Starting with -6 - A023448 n-6.
• Starting with -7 - A023449 n-7.
• Starting with -8 - A023450 n-8.
• Starting with -9 - A023451 n-9.
• Starting with -10 - A023452 n-10.
• Starting with -11 - A023453 n-11.
• Starting with -12 - A023454 n-12.
• Starting with -13 - A023455 n-13.
• Starting with -14 - A023456 n-14.
• Starting with -15 - A023457 n-15.
• Starting with -16 - A023458 n-16.
• Starting with -17 - A023459 n-17.
• Starting with -18 - A023460 n-18.
• Starting with -19 - A023461 n-19.
• Starting with -20 - A023462 n-20.
• Starting with -21 - A023463 n-21.
• Starting with -22 - A023464 n-22.
• Starting with -23 - A023465 n-23.
• Starting with -24 - A023466 n-24.
• Starting with -25 - A023467 n-25.
• Starting with -26 - A023468 n-26.
• Starting with -27 - A023469 n-27.
• Starting with -28 - A023470 n-28.
• Starting with -29 - A023471 n-29.
• Starting with -30 - A023472 n-30.
• Starting with -31 - A023473 n-31.
• Starting with -32 - A023474 n-32.
• Starting with -33 - A023475 n-33.
• Starting with -34 - A023476 n-34.
• Starting with -35 - A023477 n-35.
• Starting with -36 - A023478 n-36.
• Starting with -37 - A023479 n-37.
• Starting with -38 - A023480 n-38.
• Starting with -39 - A023481 n-39.
• Starting with -40 - A023482 n-40.
• a(n) = a(n-1) + 2.
• a(n) =2n - 1.
Sequence: 1, 3, 5, 7, 9, 11, 13,
In OEIS:
• Starting with 1 - A005408 The odd numbers: a(n) = 2n+1.
• Starting with -1 - A060747 2n-1.
• Starting with 5 - A049013 Values of n such that a regular polygon with n sides can be formed by tying knots in a strip of paper.
• Starting with 5 - A020735 Pisot sequence T(5,7).
• Starting with 7 - A020742 Pisot sequence T(7,9).
• a(n) =2n.
Sequence: 2, 4, 6, 8, 10, 12, 14,
In OEIS:
• Starting with 0 - A005843 The even numbers: a(n) = 2n.
• Starting with 2 - A087113 Number of prime divisors with multiplicity of product of terms of n-th row of A077553.
• Starting with 6 - A020739 Pisot sequence T(6,8).
• Starting with 8 - A020744 Pisot sequences P(8,10), T(8,10).
• a(n) = a(n-1) + 3.
• a(n) = 3n+1.
Sequence: 1, 4, 7, 10, 13, 16, 19, In OEIS: - A016777 3n+1; and similar to A112414 3n+7.
• a(n) = 3n+2.
Sequence: 2, 5, 8, 11, 14, 17, 20, In OEIS: - A016789 3n+2.
• a(n) = 3n.
Sequence: 3, 6, 9, 12, 15, 18, 21, In OEIS: - A008585 Multiples of 3.
• a(n) = a(n-1) + 4.
• a(n) = 4n.
Sequence: 0, 4, 8, 12, 16, 20, 24, In OEIS: - A008586 Multiples of 4.
• a(n) = 4n+1.
Sequence: 1, 5, 9, 13, 17, 21, 25, In OEIS: - A016813 4n+1; and similar to A004766 Binary expansion ends 01.
• a(n) = 4n+2.
Sequence: 2, 6, 10, 14, 18, 22, 26, In OEIS: - A016825 4n+2. Also A073760 Smallest unrelated number belonging to a term of this sequence equals four.
• a(n) = 4n+3.
Sequence: 3, 7, 11, 15, 19, 23, 27, In OEIS: - A004767 4n+3.
• a(n) = a(n-1) + 5.
• a(n) = 5n.
Sequence: 0, 5, 10, 15, 20, 25, 30, In OEIS: - A008587 Multiples of 5.
• a(n) = 5n+1.
Sequence: 1, 6, 11, 16, 21, 26, 31, In OEIS: - A016861 5n+1.
• a(n) = 5n+2.
Sequence: 2, 7, 12, 17, 22, 27, 32, In OEIS: - A016873 5n+2.
• a(n) = 5n+3.
Sequence: 3, 8, 13, 18, 23, 28, 33, In OEIS: - A016885 5n+3.
• a(n) = 5n+4.
Sequence: 4, 9, 14, 19, 24, 29, 34, In OEIS: - A016897 5n+4.
• a(n) = a(n-1) + 6.
• a(n) = 6n.
Sequence: 0, 6, 12, 18, 24, 30, 36, In OEIS: - A008588 Multiples of 6.
• a(n) = 6n+1.
Sequence: 1, 7, 13, 19, 25, 31, 37, In OEIS: - A016921 6n+1.
• a(n) = 6n+2.
Sequence: 2, 8, 14, 20, 26, 32, 38, In OEIS: - A016933 6n+2.
• a(n) = 6n+3.
Sequence: 3, 9, 15, 21, 27, 33, 39, In OEIS: - A016945 6n+3.
• a(n) = 6n+4.
Sequence: 4, 10, 16, 22, 28, 34, 40, In OEIS: - A016957 6n+4.
• a(n) = 6n+5.
Sequence: 5, 11, 17, 23, 29, 35, 41, In OEIS: - A016969 6n+5.
• a(n) = a(n-1) + 7.
• a(n) = 7n.
Sequence: 0, 7, 14, 21, 28, 35, 42, In OEIS: - A008589 Multiples of 7.
• a(n) = 7n+1.
Sequence: 1, 8, 15, 22, 29, 36, 43, In OEIS: - A016993 (7n+1).
• a(n) = 7n+2.
Sequence: 2, 9, 16, 23, 30, 37, 44, In OEIS: - A017005 7n+2.
• a(n) = 7n+3.
Sequence: 3, 10, 17, 24, 31, 38, 45, In OEIS: - A017017 7n+3.
• a(n) = 7n+4.
Sequence: 4, 11, 18, 25, 32, 39, 46, In OEIS: - A017029 7n+4.
• a(n) = 7n+5.
Sequence: 5, 12, 19, 26, 33, 40, 47, In OEIS: - A017041 (7n+5).
• a(n) = 7n+6.
Sequence: 6, 13, 20, 27, 34, 41, 48, In OEIS: - A017053 (7n+6).
• a(n) = a(n-1) + 8.
• a(n) = 8n.
Sequence: 0, 8, 16, 24, 32, 40, 48, In OEIS: - A008590 Multiples of 8.
• a(n) = 8n+1.
Sequence: 1, 9, 17, 25, 33, 41, 49, In OEIS: - A017077 (8n+1); and similar to A004768 Binary expansion ends 001.
• a(n) = 8n+2.
Sequence: 2, 10, 18, 26, 34, 42, 50, In OEIS: - A017089 8n+2.
• a(n) = 8n+3.
Sequence: 3, 11, 19, 27, 35, 43, 51, In OEIS: - A017101 8n+3; and similar to A004769 Binary expansion ends 011.
• a(n) = 8n+4.
Sequence: 4, 12, 20, 28, 36, 44, 52, In OEIS: - A017113 8n+4.
• a(n) = 8n+5.
Sequence: 5, 13, 21, 29, 37, 45, 53, In OEIS: - A004770 Numbers of form 8n+5; or, numbers whose binary expansion ends 101.
• a(n) = 8n+6.
Sequence: 6, 14, 22, 30, 38, 46, 54, In OEIS: - A017137 (8n+6).
• a(n) = 8n+7.
Sequence: 7, 15, 23, 31, 39, 47, 55, In OEIS: - A004771 a(n) = 8n+7. Or, numbers n such that binary expansion ends 111.
• a(n) = a(n-1) + 9.
• a(n) = 9n.
Sequence: 0, 9, 18, 27, 36, 45, 54, In OEIS: - A008591 Multiples of 9.
• a(n) = 9n+1.
Sequence: 1, 10, 19, 28, 37, 46, 55, In OEIS: - A017173 9n+1.
• a(n) = 9n+2.
Sequence: 2, 11, 20, 29, 38, 47, 56, In OEIS: - A017185 9n+2.
• a(n) = 9n+3.
Sequence: 3, 12, 21, 30, 39, 48, 57, In OEIS: - A017197 a(n) = 9*n + 3.
• a(n) = 9n+4.
Sequence: 4, 13, 22, 31, 40, 49, 58, In OEIS: - A017209 9n+4.
• a(n) = 9n+5.
Sequence: 5, 14, 23, 32, 41, 50, 59, In OEIS: - A017221 (9n+5).
• a(n) = 9n+6.
Sequence: 6, 15, 24, 33, 42, 51, 60, In OEIS: - A017233 9n+6.
• a(n) = 9n+7.
Sequence: 7, 16, 25, 34, 43, 52, 61, In OEIS: - A017245 (9n+7).
• a(n) = 9n+8.
Sequence: 8, 17, 26, 35, 44, 53, 62, In OEIS: - A017257 (9n+8).
• a(n) = a(n-1) + 10.
• a(n) = 10n.
Sequence: 0, 10, 20, 30, 40, 50, 60, In OEIS: - A008592 Multiples of 10.
• a(n) = 10n+1.
Sequence: 1, 11, 21, 31, 41, 51, 61, In OEIS: - A017281 10n+1.
• a(n) = 10n+2.
Sequence: 2, 12, 22, 32, 42, 52, 62, In OEIS: - A017293 10n+2.
• a(n) = 10n+3.
Sequence: 3, 13, 23, 33, 43, 53, 63, In OEIS: - A017305 10n+3.
• a(n) = 10n+4.
Sequence: 4, 14, 24, 34, 44, 54, 64, In OEIS: - A017317 10n+4.
• a(n) = 10n+5.
Sequence: 5, 15, 25, 35, 45, 55, 65, In OEIS: - A017329 10n+5.
• a(n) = 10n+6.
Sequence: 6, 16, 26, 36, 46, 56, 66, In OEIS: - A017341 (10n+6).
• a(n) = 10n+7.
Sequence: 7, 17, 27, 37, 47, 57, 67, In OEIS: - A017353 10n+7.
• a(n) = 10n+8.
Sequence: 8, 18, 28, 38, 48, 58, 68, In OEIS: - A017365 (10n+8).
• a(n) = 10n+9.
Sequence: 9, 19, 29, 39, 49, 59, 69, In OEIS: - A017377 10n+9.
• a(n) = a(n-1) + 11.
• a(n) = 11n.
Sequence: 0, 11, 22, 33, 44, 55, 66, In OEIS: - A008593 Multiples of 11.
• a(n) = 11n+1.
Sequence: 1, 12, 23, 34, 45, 56, 67, In OEIS: - A017401 11n+1.
• a(n) = 11n+2.
Sequence: 2, 13, 24, 35, 46, 57, 68, In OEIS: - A017413 11n+2.
• a(n) = 11n+3.
Sequence: 3, 14, 25, 36, 47, 58, 69, In OEIS: - A017425 11n+3.
• a(n) = 11n+4.
Sequence: 4, 15, 26, 37, 48, 59, 70, In OEIS: - A017437 11n+4.
• a(n) = 11n+5.
Sequence: 5, 16, 27, 38, 49, 60, 71, In OEIS: - A017449 11n+5.
• a(n) = 11n+6.
Sequence: 6, 17, 28, 39, 50, 61, 72, In OEIS: - A017461 (11n+6).
• a(n) = 11n+7.
Sequence: 7, 18, 29, 40, 51, 62, 73, In OEIS: - A017473 (11n+7).
• a(n) = 11n+8.
Sequence: 8, 19, 30, 41, 52, 63, 74, In OEIS: - A017485 (11n+8).
• a(n) = 11n+9.
Sequence: 9, 20, 31, 42, 53, 64, 75, In OEIS: - A017497 11n+9.
• a(n) = 11n+10.
Sequence: 10, 21, 32, 43, 54, 65, 76, In OEIS: - A017509 (11n+10).
• a(n) = a(n-1) + 12.
• a(n) = 12n.
Sequence: 0, 12, 24, 36, 48, 60, 72, In OEIS: - A008594 Multiples of 12.
• a(n) = 12n+1.
Sequence: 1, 13, 25, 37, 49, 61, 73, In OEIS: - A017533 (12n+1).
• a(n) = 12n+2.
Sequence: 2, 14, 26, 38, 50, 62, 74, In OEIS: - A017545 12n+2.
• a(n) = 12n+3.
Sequence: 3, 15, 27, 39, 51, 63, 75, In OEIS: - A017557 12n+3.
• a(n) = 12n+4.
Sequence: 4, 16, 28, 40, 52, 64, 76, In OEIS: - A017569 12n+4.
• a(n) = 12n+5.
Sequence: 5, 17, 29, 41, 53, 65, 77, In OEIS: - A017581 (12n+5).
• a(n) = 12n+7.
Sequence: 7, 19, 31, 43, 55, 67, 79, In OEIS: - A017605 12n+7.
• a(n) = 12n+8.
Sequence: 8, 20, 32, 44, 56, 68, 80, In OEIS: - A017617 (12n+8).
• a(n) = 12n+9.
Sequence: 9, 21, 33, 45, 57, 69, 81, In OEIS: - A017629 (12n+9).
• a(n) = 12n+10.
Sequence: 10, 22, 34, 46, 58, 70, 82, In OEIS: - A017641 (12n+10).
• a(n) = 12n+11.
Sequence: 11, 23, 35, 47, 59, 71, 83, In OEIS: - A017653 (12n+11).
• a(n) = a(n-1) + 13.
• a(n) = 13n.
Sequence: 0, 13, 26, 39, 52, 65, 78, In OEIS: - A008595 Multiples of 13.
• a(n) = a(n-1) + 14.
• a(n) = 14n.
Sequence: 0, 14, 28, 42, 56, 70, 84, In OEIS: - A008596 Multiples of 14.
• a(n) = a(n-1) + 15.
• a(n) = 15n.
Sequence: 0, 15, 30, 45, 60, 75, 90, In OEIS: - A008597 Multiples of 15.
• a(n) = a(n-1) + 16.
• a(n) = 16n.
Sequence: 0, 16, 32, 48, 64, 80, 96, In OEIS: - A008598 Multiples of 16.
• a(n) = 16n+4.
Sequence: 4, 20, 36, 52, 68, 84, 100, In OEIS: - A119413 16*n-12.
• a(n) = 16n+8.
Sequence: 8, 24, 40, 56, 72, 88, 104, In OEIS: - A051062 16n+8.
• a(n) = 16n+11.
Sequence: 11, 27, 43, 59, 75, 91, 107, In OEIS: - A106839 Numbers congruent to 11 mod 16.
• a(n) = 16n+12.
Sequence: 12, 28, 44, 60, 76, 92, 108, In OEIS: - A098502 16n - 4.
• a(n) = 16n+13.
Sequence: 13, 29, 45, 61, 77, 93, 109, In OEIS: - A082285 Solutions to 7^x+11^x == 13 mod 17.
• a(n) = 16n+15.
Sequence: 15, 31, 47, 63, 79, 95, 111, In OEIS: - A125169 16n+15.
• a(n) = a(n-1) + 17.
• a(n) = 17n.
Sequence: 0, 17, 34, 51, 68, 85, 102, In OEIS: - A008599 Multiples of 17.
• a(n) = a(n-1) + 18.
• a(n) = 18n.
Sequence: 0, 18, 36, 54, 72, 90, 108, In OEIS: - A008600 Multiples of 18.
• a(n) = 18n+10.
Sequence: 10, 28, 46, 64, 82, 100, 118, In OEIS: - A082286 Numbers of the form 18n-8.
• a(n) = 18n+14.
Sequence: 32, 50, 68, 86, 104, 122, 140, In OEIS: - A099048 Number of 5 X n 0-1 matrices avoiding simultaneously the right angled numbered polyomino patterns (ranpp) (00;1), (01;0), (10;0), and (11;0). An occurrence of a ranpp (xy;z) in a matrix A=(a(i,j)) is a triple (a(i1,j1), a(i1,j2), a(i2,j1)) where i1<i2, j1<j2, and these elements are in the same relative order as those in the triple (x,y,z). In general, the number of m X n 0-1 matrices in question is given by (n+1)2^(m-1)+2(n-1).
• a(n) = a(n-1) + 19.
• a(n) = 19n.
Sequence: 0, 19, 38, 57, 76, 95, 114, In OEIS: - A008601 Multiples of 19.
• a(n) = a(n-1) + 20.
• a(n) = 20n.
Sequence: 0, 20, 40, 60, 80, 100, 120, In OEIS: - A008602 Multiples of 20.
• a(n) = a(n-1) + 21.
• a(n) = 21n.
Sequence: 0, 21, 42, 63, 84, 105, 126, In OEIS: - A008603 Multiples of 21.
• a(n) = a(n-1) + 22.
• a(n) = 22n.
Sequence: 0, 22, 44, 66, 88, 110, 132, In OEIS: - A008604 Multiples of 22.
• a(n) = a(n-1) + 23.
• a(n) = 23n.
Sequence: 0, 23, 46, 69, 92, 115, 138, In OEIS: - A008605 Multiples of 23.
• a(n) = a(n-1) + 24.
• a(n) = 24n.
Sequence: 0, 24, 48, 72, 96, 120, 144, In OEIS: - A008606 Multiples of 24.
• a(n) = 24n+1.
Sequence: 1, 25, 49, 73, 97, 121, 145, In OEIS: - A103214 24n + 1.
• a(n) = 24n+12.
Sequence: 12, 36, 60, 84, 108, 132, 156, In OEIS: - A073762 Smallest unrelated number belonging to a term of this sequence equals 8.
• a(n) = 24n+21.
Sequence: 21, 45, 69, 93, 117, 141, 165, In OEIS: - A102603 24n + 21.
• a(n) = a(n-1) + 25.
• a(n) = 25n.
Sequence: 0, 25, 50, 75, 100, 125, 150, In OEIS: - A008607 Multiples of 25.
• a(n) = a(n-1) + 26.
• a(n) = 26n+20.
Sequence: 72, 98, 124, 150, 176, 202, 228, In OEIS: - A099943 Number of 5 X n binary matrices avoiding simultaneously the right angled numbered polyomino patterns (ranpp) (00;1), (01,1), and (11;0). An occurrence of a ranpp (xy;z) in a matrix A=(a(i,j)) is a triple (a(i1,j1), a(i1,j2), a(i2,j1)) where i1<i2, j1<j2, and these elements are in the same relative order as those in the triple (x,y,z). In general, the number of m X n 0-1 matrices in question is given by (n+2)*2^(m-1)+2*m*(n-1)-2 for m>1 and n>1.
• a(n) = a(n-1) + 27.
• a(n) = 27n+18.
Sequence: 18, 45, 72, 99, 126, 153, 180, In OEIS: - A124388 Second differences of dodecahedral numbers (A006566).
• a(n) = a(n-1) + 30.
• a(n) = 30n+13.
Sequence: 13, 43, 73, 103, 133, 163, 193, In OEIS: - A082369 Solutions to 19^x+23^x == 29 mod 31.
• a(n) = a(n-1) + 37.
• a(n) = 37n.
Sequence: 0, 37, 74, 111, 148, 185, 222, In OEIS: - A085959 Multiples of 37.
• a(n) = a(n-1) + 50.
• a(n) = 50n+20.
Sequence: 20, 70, 120, 170, 220, 270, 320, In OEIS: - A053741 Sum of even numbers in range 10n to 10n+9.
• a(n) = 50n+25.
Sequence: 25, 75, 125, 175, 225, 275, 325, In OEIS: - A053742 Sum of odd numbers in range 10n to 10n+9.
• a(n) = a(n-1) + 75.
• a(n) = 75n+3.
Sequence: 3, 78, 153, 228, 303, 378, In OEIS: - A097802 3(25*n + 1)
• a(n) = a(n-1) + 97.
• a(n) = 97n+101.
Sequence: 101, 198, 295, 392, 489, 586, In OEIS: - A100775 a(n) = 97*n + 101.
• a(n) = a(n-1) + 100.
• a(n) = 100n+4.
Sequence: 104, 204, 304, 404, 504, 604, In OEIS: - A102439 Numbers of the form s04 divisible by 4.
• a(n) = 100n+36.
Sequence: 36, 136, 236, 336, 436, 536, In OEIS: - A067865 Numbers n such that n and 2^n end with the same two digits.
• a(n) = 100n+44.
Sequence: 44, 144, 244, 344, 444, 544, In OEIS: - A102438 Numbers of the form s44 that are divisible by 4.
• a(n) = 100n+45.
Sequence: 45, 145, 245, 345, 445, 545, In OEIS: - A053743 Sum of numbers in range 10n to 10n+9.
• a(n) = 100n+87.
Sequence: 87, 187, 287, 387, 487, 587, In OEIS: - A067749 Numbers n such that n and 3^n end with the same two digits.
• a(n) = a(n-1) + 101.
• a(n) = 101n+1.
Sequence: 1, 102, 203, 304, 405, 506, In OEIS: - A078787 101*n + 1.
• a(n) = a(n-1) + 180.
• a(n) = 180n.
Sequence: 180, 360, 540, 720, 900, In OEIS: - A066164 Sum of interior angles in an n-sided polygon in degrees.
• a(n) = a(n-1) + 200.
• a(n) = 180n.
Sequence: 200, 400, 600, 800, 1000, In OEIS: - A117412 Sum of the interior angles of an n-sided polygon, in gradians.
• a(n) = a(n-1) + 512.
• a(n) = 512n+1.
Sequence: 1, 513, 1025, 1537, 2049, 2561, In OEIS: - A076338 512*n + 1.
• a(n) = a(n-1) + 720.
• a(n) = 720n+1800.
Sequence: 2520, 3240, 3960, 4680, 5400, In OEIS: - A069476 First differences of A069475, successive differences of (n+1)^6-n^6.
• a(n) = a(n-1) + 840.
• a(n) = 840n+423.
Sequence: 423, 1263, 2103, 2943, 3783, 4623, In OEIS: - A096024 Numbers n such that (n+j) mod (2+j) = 1 for j from 0 to 5 and (n+6) mod 8 <> 1.
• a(n) = a(n-1) + 997.
• a(n) = 997n+1009.
Sequence: 1009, 2006, 3003, 4000, 4997, In OEIS: - A100776 a(n) = 997 * n + 1009.
• a(n) = a(n-1) + 1000.
• a(n) = 1000n+736.
Sequence: 736, 1736, 2736, 3736, 4736, In OEIS: - A067866 Numbers n such that n and 2^n end with the same three digits.
• a(n) = a(n-1) + 3018.
• a(n) = 3018n.
Sequence: 3018, 6036, 9054, 12072, 15090, In OEIS: - A086746 Multiples of 3018.
• a(n) = a(n-1) + 3600.
• a(n) = 3600n.
Sequence: 3600, 7200, 10800, 14400, 18000, In OEIS: - A096472 Numbers containing Pythagorean triples in their divisor set.
• a(n) = a(n-1) + 10000.
• a(n) = 10000n+2468.
Sequence: 2468, 12468, 22468, 32468, 42468, In OEIS: - A102689 Numbers of the form s2468 divisible by 2 and 4.
• a(n) = a(n-1) + 9973.
• a(n) = 9973n+10007.
Sequence: 10007, 19980, 29953, 39926, 49899, In OEIS: - A101442 a(n) = 9973*n + 10007.
• a(n) = a(n-1) + 10000.
• a(n) = 10000n+8736.
Sequence: 8736, 18736, 28736, 38736, 48736, In OEIS: - A067867 Numbers n such that n and 2^n end with the same 4 digits.
• a(n) = a(n-1) + 21460.
• a(n) = 21460n+1123.
Sequence: 1123, 22583, 44043, 65503, 86963, In OEIS: - A069984 1123+21460n.
• a(n) = a(n-1) + 100000.
• a(n) = 100000n+48736.
Sequence: 48736, 148736, 248736, 348736, In OEIS: - A067869 Numbers n such that n and 2^n end with the same 5 digits.
• a(n) = a(n-1) + 142857.
• a(n) = 142857n.
Sequence: 142857, 285714, 428571, 571428, 714285, In OEIS: - A101202 Multiples of 142857.
• a(n) = a(n-1) + 1968751.
• a(n) = 1968751n+533360.
Sequence: 533360, 2502111, 4470862, 6439613, In OEIS: - A010037 G.c.d. (n^5 + 5, (n+1)^5 + 5) more than 1.
• a(n) = a(n-1) + 11184810.
• a(n) = 11184810n+7629217.
Sequence: 7629217, 18814027, 29998837, In OEIS: - A080340 First known infinite sequence containing no odd integer of the form 2^m+p (p prime).
• a(n) = a(n-1) + 12345679.
• a(n) = 12345679n.
Sequence: 0, 12345679, 24691358, 37037037, In OEIS: - A070189 12345679*n.
• a(n) = a(n-1) + 123456789.
• a(n) = 123456789n.
Sequence: 123456789, 246913578, 370370367, In OEIS: - A053654 Multiples of 123456789.
• a(n) = a(n-1) + 1757711340.
• a(n) = 1757711340n + 242.
Sequence: 242, 1757711582, 3515422922, 5273134262, In OEIS: - A055554 An arithmetic progression each term of which is followed by at least 4 non-square-free consecutive integers.
• a(n) = a(n-1) + 8936582237915716659950962253358945635793453256935559.
• a(n) = 8936582237915716659950962253358945635793453256935559n - 512149312322827330662764931050044963334032796143126.
Sequence: 8424432925592889329288197322308900672459420460792433, 17361015163508605989239159575667846308252873717727992, In OEIS: - A010034 Numbers n such that g.c.d. (n^17 + 9, (n+1)^17 + 9) > 1.

### a(n) = a(n-1). Constants.

Sequences:

• a(n) = 0.
Sequence: 0, 0, 0, 0, 0,
In OEIS: - A000004 The zero sequence.
• a(n) = 1.
Sequence: 1, 1, 1, 1, 1,
In OEIS: - A000012 The simplest sequence of positive numbers: the all 1's sequence.
• a(n) = 2.
Sequence: 2, 2, 2, 2, 2,
In OEIS: - A007395 The all 2's sequence.
• a(n) = 3.
Sequence: 3, 3, 3, 3, 3,
In OEIS: - A010701 Constant sequence.
• a(n) = 4.
Sequence: 4, 4, 4, 4, 4,
In OEIS: - A010709 Constant sequence.
• a(n) = 5.
Sequence: 5, 5, 5, 5, 5, In OEIS: - A010716 Constant sequence.
• a(n) = 6.
Sequence: 6, 6, 6, 6, 6, In OEIS: - A010722 Constant sequence.
• a(n) = 7.
Sequence: 7, 7, 7, 7, 7, In OEIS: - A010727 Constant sequence.
• a(n) = 8.
Sequence: 8, 8, 8, 8, 8, In OEIS: - A010731 Constant sequence.
• a(n) = 9.
Sequence: 9, 9, 9, 9, 9, In OEIS: - A010734 Constant sequence.
• a(n) = 10.
Sequence: 10, 10, 10, 10, 10, In OEIS: - A010692 Constant sequence.
• a(n) = 11.
Sequence: 11, 11, 11, 11, 11, In OEIS: - A010850 Constant sequence.
• a(n) = 12.
Sequence: 12, 12, 12, 12, 12, In OEIS: - A010851 Constant sequence.
• a(n) = 13.
Sequence: 13, 13, 13, 13, 13, In OEIS: - A010852 Constant sequence.
• a(n) = 14.
Sequence: 14, 14, 14, 14, 14, In OEIS: - A010853 Constant sequence.
• a(n) = 15.
Sequence: 15, 15, 15, 15, 15, In OEIS: - A010854 Constant sequence.
• a(n) = 16.
Sequence: 16, 16, 16, 16, 16, In OEIS: - A010855 Constant sequence.
• a(n) = 17.
Sequence: 17, 17, 17, 17, 17, In OEIS: - A010856 Constant sequence.
• a(n) = 18.
Sequence: 18, 18, 18, 18, 18, In OEIS: - A010857 Constant sequence.
• a(n) = 19.
Sequence: 19, 19, 19, 19, 19, In OEIS: - A010858 Constant sequence.
• a(n) = 20.
Sequence: 20, 20, 20, 20, 20, In OEIS: - A010859 Constant sequence.
• a(n) = 21. In OEIS: - A010860.
• a(n) = 22. In OEIS: - A010861.
• a(n) = 23. In OEIS: - A010862.
• a(n) = 24. In OEIS: - A010863.
• a(n) = 25. In OEIS: - A010864.
• a(n) = 26. In OEIS: - A010865.
• a(n) = 27. In OEIS: - A010866.
• a(n) = 28. In OEIS: - A010867.
• a(n) = 29. In OEIS: - A010868.
• a(n) = 30. In OEIS: - A010869.
• a(n) = 31. In OEIS: - A010870.
• a(n) = 32. In OEIS: - A010871.

Last revised March 2007