Non Recursions

These are sequences whose initially available terms passed tests for simple recurrences, although their definitions did not establish those recurrences. The formulas below record apparent recurrences and need not hold for all terms. Comments distinguish failures of the displayed formulas from finite lists, obsolete entries, and confirmed recurrences. A failure of the displayed formula does not rule out other recurrences. See the main page Recursive Sequences.

• Interesting sequences and their status.
• a(n) = d * a(n-1) - a(n-2).
• a(n) = d * a(n-1) + d * a(n-2).
• a(n) = a(n-1) + a(n-2).
• a(n) = d * a(n-1) + a(n-2).
• a(n) = d * a(n-1). Geometric progressions.
• a(n) = a(n-1) + d. Arithmetic progressions.
• a(n) = a(n-1). Constants.
• Comments.

This page was last fully synchronized with OEIS in January 2007. The apparent recurrences are indexed from 0, independently of the parameters in the sequence descriptions; current OEIS entries may use different offsets. Some descriptions, links, and comments have since been corrected.

Interesting Sequences.

Sequences:

• A123464 Recurrence: a(n) = 2*a(n-1), a(0) = 1.
  Number of unlabeled threshold graphs on n+1 nodes (with the indexing used here).
  The displayed recurrence holds. This obsolete entry is a duplicate of A011782, with the empty graph omitted.
• A070300 Recurrence: a(n) = a(n-1) + 3, a(0) = 4.
  Minimal number of 0's in a 2k X 2k (0,1) matrix that contains no k X k submatrix of 1's, with k = n+1.
  The displayed recurrence holds: the minimum is 3k+1, by a theorem of Griggs and Ouyang (see Theorem 1.1 in Griggs and Ho).
• A085805 Apparent recurrence: a(n) = a(n-1) + 16, a(0) = 4.
  Numbers n such that the permanent of the character table of the dihedral group D_n is not zero.
  The displayed recurrence is conjectural in the OEIS entry.
• A114142 Recurrence: a(n) = a(n-1) + 1, a(0) = 2.
  Possible positive sums of the final scores of a completed American football game.
  The displayed recurrence holds. OEIS now identifies this entry as a duplicate of A020725.
• A114143 Recurrence: a(n) = a(n-1) + 1, a(0) = 4.
  Possible sums of the final scores of a completed American football game in which both teams score.
  The displayed recurrence holds.
• A118759 Apparent recurrence: a(n) = a(n-1) + 1, a(0) = 0.
  A118757(A118757(n)).
  The displayed recurrence first fails at n = 100, where a(100) = 189.
• A118760 Apparent recurrence: a(n) = a(n-1) + 1, a(0) = 0.
  A118758(A118758(n)).
  The displayed recurrence first fails at n = 100, where a(100) = 118.

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

Sequences:

• a(n) = 2a(n-1) - a(n-2). For d = 2 see arithmetic progressions.
• A011783 Apparent recurrence: a(n) = 3*a(n-1) - a(n-2), a(0) = 1, a(1) = 1.
  Duplicate of A001519. (dead sequence)
• A077461 Apparent recurrence: a(n) = 6*a(n-1) - a(n-2), a(0) = 2, a(1) = 14.
  Same as A077444. (dead sequence)
• A001075 Recurrence: a(n) = 4*a(n-1) - a(n-2), a(0) = 1, a(1) = 2.
  The displayed recurrence holds. The original page associated this formula with A087096, whose current OEIS entry is unrelated.

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

Sequences:

• a(n) = a(n-1) + a(n-2). For d = 1 see a(n) = a(n-1) + a(n-2).
• A085481 Apparent recurrence: a(n) = 3*a(n-1) + 3*a(n-2), a(0) = 1, a(1) = 3.
  Same as A030195. (dead sequence)

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

Sequences:

• A077373 Apparent recurrence: a(n) = a(n-1)+a(n-2), a(0) = 0, a(1) = 1.
  Fibonacci numbers whose external digits as well as internal digits form a Fibonacci number.
  The displayed recurrence fails: after 89 it predicts 144, whose outer digits form 14 and whose inner digit is 4; neither is a Fibonacci number.

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

Sequences:

• a(n) = a(n-1) + a(n-2). For d = 1 see a(n) = a(n-1) + a(n-2).
• A048624 Apparent recurrence: a(n) = 2*a(n-1)+a(n-2), a(0) = 2, a(1) = 5.
  Essentially a duplicate of A000129. (dead sequence)
• A084133 Apparent recurrence: a(n) = 6*a(n-1)+a(n-2), a(0) = 1, a(1) = 3.
  Duplicate of A005667. (dead sequence)

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

Sequences:

• a(n) = a(n-1). For d = 1 see constants.
• A025489 Apparent recurrence: a(n) = 2*a(n-1), a(0) = 2.
  Numbers on backgammon doubling cube.
  Finite list; the displayed formula does not specify where the list stops.
• A060365 Apparent recurrence: a(n) = 1000*a(n-1), a(0) = 1000.
  Powers of one thousand named by single words in the American English list used for this entry, starting with thousand.
  Finite list; the displayed formula does not specify where the list stops.
• A060366 Apparent recurrence: a(n) = 1000*a(n-1), a(0) = 1.
  Powers of one thousand named by single words in the dated British English convention used for this entry, including one and extending the -ard names beyond milliard.
  Finite list; the displayed formula does not specify where the list stops.
• A067482 Apparent recurrence: a(n) = 1024*a(n-1), a(0) = 4.
  Powers of 4 with initial digit 4.
  The displayed recurrence fails at n = 10: after 4⁴⁶ it predicts 4⁵¹, whose initial digit is 5.
• A067484 Apparent recurrence: a(n) = 10077696*a(n-1), a(0) = 6.
  Powers of 6 with initial digit 6.
  The displayed recurrence fails at n = 20: after 6¹⁷² it predicts 6¹⁸¹, whose initial digit is 7.
• A121499 Apparent recurrence: a(n) = 841*a(n-1), a(0) = 1.
  Denominators of partial alternating sums of Catalan numbers scaled by powers of 1/(29^2) = 1/841.
  The displayed recurrence fails. Comment by Max Alekseyev: It first fails at n = 15, when the Catalan number C(15) is divisible by 29.
• A123464 Recurrence: a(n) = 2*a(n-1), a(0) = 1.
  Number of unlabeled threshold graphs on n+1 nodes (with the indexing used here).
  The displayed recurrence holds. This obsolete entry is a duplicate of A011782, with the empty graph omitted.
• A125581 Apparent recurrence: a(n) = 11*a(n-1), a(0) = 77.
  Numbers n such that n does not divide the denominator of the n-th harmonic number nor the denominator of the n-th alternating harmonic number.
  The displayed recurrence fails. Comment by Max Alekseyev: While A125581 indeed contains the geometric progression 7*11ⁿ as a subsequence, it also contains other geometric progressions such as: 546*1093ⁿ, 1092*1093ⁿ, 1755*3511ⁿ, 3510*3511ⁿ, and 4896*5557ⁿ, with n ≥ 1 in each progression.

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

In the floor, nearest-integer, and ceiling sequences below, tau = (1 + sqrt(5))/2 is the golden ratio. Each of these apparent arithmetic progressions eventually fails, because the corresponding power of tau is irrational. For example, A004924 first fails at n = 77, A004944 at n = 39, and A004963 at n = 47.

Sequences:

• a(n) = a(n-1). For d = 0 see constants.
• A004924 Apparent recurrence: a(n) = a(n-1) + 76, a(0) = 0.
  Floor of n*tau^9.
• A004926 Apparent recurrence: a(n) = a(n-1) + 199, a(0) = 0.
  Floor of n*tau^11.
• A004928 Apparent recurrence: a(n) = a(n-1) + 521, a(0) = 0.
  Floor of n*tau^13.
• A004930 Apparent recurrence: a(n) = a(n-1) + 1364, a(0) = 0.
  Floor of n*tau^15.
• A004932 Apparent recurrence: a(n) = a(n-1) + 3571, a(0) = 0.
  Floor of n*tau^17.
• A004934 Apparent recurrence: a(n) = a(n-1) + 9349, a(0) = 0.
  Floor of n*tau^19.
• A004944 Apparent recurrence: a(n) = a(n-1) + 76, a(0) = 0.
  Nearest integer to n*tau^9.
• A004945 Apparent recurrence: a(n) = a(n-1) + 123, a(0) = 0.
  Nearest integer to n*tau^10.
• A004946 Apparent recurrence: a(n) = a(n-1) + 199, a(0) = 0.
  Nearest integer to n*tau^11.
• A004947 Apparent recurrence: a(n) = a(n-1) + 322, a(0) = 0.
  Nearest integer to n*tau^12.
• A004948 Apparent recurrence: a(n) = a(n-1) + 521, a(0) = 0.
  Nearest integer to n*tau^13.
• A004949 Apparent recurrence: a(n) = a(n-1) + 843, a(0) = 0.
  Nearest integer to n*tau^14.
• A004950 Apparent recurrence: a(n) = a(n-1) + 1364, a(0) = 0.
  Nearest integer to n*tau^15.
• A004951 Apparent recurrence: a(n) = a(n-1) + 2207, a(0) = 0.
  Nearest integer to n*tau^16.
• A004952 Apparent recurrence: a(n) = a(n-1) + 3571, a(0) = 0.
  Nearest integer to n*tau^17.
• A004953 Apparent recurrence: a(n) = a(n-1) + 5778, a(0) = 0.
  Nearest integer to n*tau^18.
• A004954 Apparent recurrence: a(n) = a(n-1) + 9349, a(0) = 0.
  Nearest integer to n*tau^19.
• A004955 Apparent recurrence: a(n) = a(n-1) + 15127, a(0) = 0.
  Nearest integer to n*tau^20.
• A004963 Apparent recurrence: a(n) = a(n-1) + 47, a(0) = 0.
  Ceiling of n*tau^8.
• A004965 Apparent recurrence: a(n) = a(n-1) + 123, a(0) = 0.
  Ceiling of n*tau^10.
• A004967 Apparent recurrence: a(n) = a(n-1) + 322, a(0) = 0.
  Ceiling of n*tau^12.
• A004969 Apparent recurrence: a(n) = a(n-1) + 843, a(0) = 0.
  Ceiling of n*tau^14.
• A004971 Apparent recurrence: a(n) = a(n-1) + 2207, a(0) = 0.
  Ceiling of n*tau^16.
• A004973 Apparent recurrence: a(n) = a(n-1) + 5778, a(0) = 0.
  Ceiling of n*tau^18.
• A004975 Apparent recurrence: a(n) = a(n-1) + 15127, a(0) = 0.
  Ceiling of n*tau^20.
• A008553 Apparent recurrence: a(n) = a(n-1) + 1, a(0) = 20.
  Numbers whose American English names contain the letter "y".
  The displayed recurrence first fails at n = 80: a(79) = 99 and a(80) = 120. None of the numbers from 100 through 119 has the letter "y" in its name.
• A017149 Apparent recurrence: a(n) = a(n-1) + 8, a(0) = 7.
  Same as A004771. (dead sequence)
• A031193 Apparent recurrence: a(n) = a(n-1) + 3, a(0) = 3.
  Numbers having period-22 5-digitized sequences.
  The displayed recurrence fails.
• A032614 Apparent recurrence: a(n) = a(n-1) + 101, a(0) = 110.
  Concatenation of n and n + 9 or {n,n+9}.
  The displayed recurrence fails.
• A033168 Apparent recurrence: a(n) = a(n-1) + 210, a(0) = 199.
  Longest arithmetic progression of primes with difference 210 and minimal initial term.
  Finite list; the displayed formula does not specify where the list stops.
• A033290 Apparent recurrence: a(n) = a(n-1) + 210, a(0) = 100996972469714247637786655587969840329509324689190041803603417758904341703348882159067229719.
  Ten consecutive primes in arithmetic progression.
  Finite list; the displayed formula does not specify where the list stops.
• A044138 Apparent recurrence: a(n) = a(n-1) + 49, a(0) = 49.
  Numbers n such that string 0,0 occurs in the base 7 representation of n but not of n-1.
  The displayed recurrence fails. See Comment.
• A044179 Apparent recurrence: a(n) = a(n-1) + 49, a(0) = 41.
  Numbers n such that string 5,6 occurs in the base 7 representation of n but not of n-1.
  The displayed recurrence fails. See Comment.
• A044187 Apparent recurrence: a(n) = a(n-1) + 64, a(0) = 64.
  Numbers n such that string 0,0 occurs in the base 8 representation of n but not of n-1.
  The displayed recurrence fails. See Comment.
• A044242 Apparent recurrence: a(n) = a(n-1) + 64, a(0) = 55.
  Numbers n such that string 6,7 occurs in the base 8 representation of n but not of n-1.
  The displayed recurrence fails. See Comment.
• A044251 Apparent recurrence: a(n) = a(n-1) + 81, a(0) = 81.
  Numbers n such that string 0,0 occurs in the base 9 representation of n but not of n-1.
  The displayed recurrence fails. See Comment.
• A044322 Apparent recurrence: a(n) = a(n-1) + 81, a(0) = 71.
  Numbers n such that string 7,8 occurs in the base 9 representation of n but not of n-1.
  The displayed recurrence fails. See Comment.
• A044332 Apparent recurrence: a(n) = a(n-1) + 100, a(0) = 100.
  Numbers n such that string 0,0 occurs in the base 10 representation of n but not of n-1.
  The displayed recurrence fails. See Comment.
• A044421 Apparent recurrence: a(n) = a(n-1) + 100, a(0) = 89.
  Numbers n such that string 8,9 occurs in the base 10 representation of n but not of n-1.
  The displayed recurrence fails. See Comment.
• A044567 Apparent recurrence: a(n) = a(n-1) + 49, a(0) = 48.
  Numbers n such that string 6,6 occurs in the base 7 representation of n but not of n+1.
  The displayed recurrence fails. See Comment.
• A044631 Apparent recurrence: a(n) = a(n-1) + 64, a(0) = 63.
  Numbers n such that string 7,7 occurs in the base 8 representation of n but not of n+1.
  The displayed recurrence fails. See Comment.
• A044712 Apparent recurrence: a(n) = a(n-1) + 81, a(0) = 80.
  Numbers n such that string 8,8 occurs in the base 9 representation of n but not of n+1.
  The displayed recurrence fails. See Comment.
• A044812 Apparent recurrence: a(n) = a(n-1) + 100, a(0) = 99.
  Numbers n such that string 9,9 occurs in the base 10 representation of n but not of n+1.
  The displayed recurrence fails. See Comment.
• A046050 Apparent recurrence: a(n) = a(n-1) + 80, a(0) = 79.
  Sum of 19 but no fewer nonzero fourth powers.
  Finite list; the displayed formula does not specify where the list stops.
• A047738 Apparent recurrence: a(n) = a(n-1) + 1, a(0) = 179210312.
  Earliest sequence of 4 consecutive economical numbers.
  Finite list; the displayed formula does not specify where the list stops.
• A050518 Apparent recurrence: a(n) = a(n-1) + 583200, a(0) = 583200.
  Starting values of arithmetic progressions of at least 6 terms on which Euler's totient function phi is constant.
  The displayed recurrence fails. With the indexing used here, the first failure is a(112) = 65621220, instead of 113*583200 = 65901600 (Robert Israel, OEIS).
• A050519 Apparent recurrence: a(n) = a(n-1) + 30, a(0) = 30.
  Increments of the arithmetic progressions in A050518.
  The displayed recurrence first fails at n = 112, where a(112) = 210 instead of 3390.
• A058908 Apparent recurrence: a(n) = a(n-1) + 9876543210, a(0) = 5077.
  Six prime numbers in arithmetical progression with a common difference of 9876543210.
  Finite list; the displayed formula does not specify where the list stops.
• A059558 Apparent recurrence: a(n) = a(n-1) + 4, a(0) = 4.
  Beatty sequence for 1+1/gamma^2, where gamma is the Euler-Mascheroni constant.
  The displayed recurrence fails. Comment by Max Alekseyev: The first failure is 2861, at index 715 in OEIS, or a(714) = 2861 with the indexing used here.
• A069782 Apparent recurrence: a(n) = a(n-1) + 1, a(0) = 1.
  Numbers n such that g[n] := GCD[d[n^3],d[n]] = 2^w for some w. The first missing integer is 432 (See in A069781).
  The displayed recurrence fails.
• A070300 Recurrence: a(n) = a(n-1) + 3, a(0) = 4.
  Minimal number of 0's in a 2k X 2k (0,1) matrix that contains no k X k submatrix of 1's, with k = n+1.
  The displayed recurrence holds: the minimum is 3k+1, by a theorem of Griggs and Ouyang (see Theorem 1.1 in Griggs and Ho).
• A074337 Apparent recurrence: a(n) = a(n-1) + 9922782870, a(0) = 107928278317.
  18 primes in arithmetic progression.
  Finite list; the displayed formula does not specify where the list stops.
• A081734 Apparent recurrence: a(n) = a(n-1) + 30, a(0) = 121174811.
  First and smallest sequence of 6 consecutive primes in arithmetic progression.
  Finite list; the displayed formula does not specify where the list stops.
• A082221 Apparent recurrence: a(n) = a(n-1) + 6, a(0) = 1.
  In the following square array, the smallest unused positive integers satisfying the conditions are entered in this order: a(1,1),a(1,2),a(2,1),a(3,1),a(2,2),a(1,3),a(1,4),a(2,3),a(3,2),a(4,1),a(5,1),a(4,2),... such that every n-th partial sum of a row or a column is a multiple of n. 1 3 2 10 19 25... 5 7 12 16 15... 6 8 13 37... 4 14 21... 9 23... 11... ... This sequence is the main diagonal of the array A082218.
  The displayed recurrence fails at n = 3, where a(3) = 29 instead of 19. More terms were calculated by Max Alekseyev.
• A082249 Apparent recurrence: a(n) = a(n-1) + 7070707070707, a(0) = 22212019181716.
  Multiples of 7 formed by concatenating seven consecutive positive integers in decreasing order.
  The displayed recurrence fails. Comment by Max Alekseyev: The term 108107106105104103102 does not lie in the apparent arithmetic progression.
• A082946 Apparent recurrence: a(n) = a(n-1) + 101, a(0) = 111.
  Palindromes satisfying A082945.
  The displayed recurrence fails. Comment by Max Alekseyev: the displayed recurrence predicts 919+101=1020 after 919, but 1020 is not a palindrome.
• A085805 Apparent recurrence: a(n) = a(n-1) + 16, a(0) = 4.
  Numbers n such that the permanent of the character table of the dihedral group D_n is not zero.
  The displayed recurrence is conjectural in the OEIS entry.
• A088475 Apparent recurrence: a(n) = a(n-1) + 1, a(0) = 10.
  Numbers n such that the dismal sum of the distinct dismal prime divisors of n is ≥ n.
  The displayed recurrence fails.
• A088480 Apparent recurrence: a(n) = a(n-1) + 1, a(0) = 1.
  Numbers n such that the dismal product of the distinct dismal prime divisors of n is ≥ n.
  The displayed recurrence fails. Complement of A088477
• A096582 Apparent recurrence: a(n) = a(n-1) - 1, a(0) = 100.
  From the "100 Green Bottles" song.
  Finite list; the displayed formula does not specify where the list stops.
• A103303 Apparent recurrence: a(n) = a(n-1) + 1, a(0) = 0.
  Complete list of digits used in the counting numbers (in base 10). Also known as the "arabic numerals".
  Finite list; the displayed formula does not specify where the list stops.
• A104340 Apparent recurrence: a(n) = a(n-1) + 11, a(0) = 12.
  Numbers n such that (digital reversal of n) - n = 9.
  Finite list; the displayed formula does not specify where the list stops.
• A104341 Apparent recurrence: a(n) = a(n-1) + 11, a(0) = 10.
  Numbers n such that n -(digital reversal of n) = 9.
  Finite list; the displayed formula does not specify where the list stops.
• A104342 Apparent recurrence: a(n) = a(n-1) + 11, a(0) = 13.
  Numbers n such that (digital reversal of n) - n = 18.
  Finite list; the displayed formula does not specify where the list stops.
• A107843 Apparent recurrence: a(n) = a(n-1) - 2, a(0) = 201.
  Number of iterations of McCarthy 91 Function until it terminates.
  The displayed recurrence fails.
• A109065 Apparent recurrence: a(n) = a(n-1) - 1, a(0) = 12.
  Numerator of the fraction due in month n of the total interest for a one-year installment loan based on the Rule of 78s (Each denominator is 78).
  Finite list; the displayed formula does not specify where the list stops.
• A109632 Apparent recurrence: a(n) = a(n-1) + 300, a(0) = 200.
  In the game of bridge, a(n) is the penalty for going down n tricks in a vulnerable, doubled contract.
  Finite list; the displayed formula does not specify where the list stops.
• A112821 Apparent recurrence: a(n) = a(n-1) + 1, a(0) = 343.
  Numbers n such that LCM(1,2,3,...,n)/19 equals the denominator of the n-th harmonic number H(n).
  The displayed recurrence fails. Comment by Max Alekseyev: 361 does not belong to A112821.
• A114142 Recurrence: a(n) = a(n-1) + 1, a(0) = 2.
  Possible positive sums of the final scores of a completed American football game.
  The displayed recurrence holds. OEIS now identifies this entry as a duplicate of A020725.
• A114143 Recurrence: a(n) = a(n-1) + 1, a(0) = 4.
  Possible sums of the final scores of a completed American football game in which both teams score.
  The displayed recurrence holds.
• A115020 Apparent recurrence: a(n) = a(n-1) - 7, a(0) = 100.
  Count backwards from 100 in steps of 7.
  Finite list; the displayed formula does not specify where the list stops.
• A115536 Apparent recurrence: a(n) = a(n-1) + 9434, a(0) = 160378.
  Numbers n such that the square of n is the concatenation of two numbers m and 4*m, in either order.
  The displayed recurrence predicts 500002, but its square 250002000004 does not satisfy the concatenation condition.
• A115548 Apparent recurrence: a(n) = a(n-1) + 30434782608695652173913043478260869565217391304347826087, a(0) = 91304347826086956521739130434782608695652173913043478261.
  Numbers n such that the square of n is the concatenation of two numbers m and 7*m, in either order.
  The displayed recurrence predicts 3*a(0) at index 6, but its square does not satisfy the concatenation condition.
• A118759 Apparent recurrence: a(n) = a(n-1) + 1, a(0) = 0.
  A118757(A118757(n)).
  The displayed recurrence first fails at n = 100, where a(100) = 189.
• A118760 Apparent recurrence: a(n) = a(n-1) + 1, a(0) = 0.
  A118758(A118758(n)).
  The displayed recurrence first fails at n = 100, where a(100) = 118.
• Historical factorial entry, originally listed here as A120420. Apparent recurrence: a(n) = a(n-1) + 1, a(0) = 2.
  Integers n ≥ 2 such that n! is highly composite (in the sense of A002182).
  This is the finite list 2, 3, 4, 5, 6, 7. The current entry A120420 concerns a different sequence.
• A121377 Apparent recurrence: a(n) = a(n-1) + 1, a(0) = 48.
  ASCII codes for decimal digits.
  Finite list; the displayed formula does not specify where the list stops.
• A121378 Apparent recurrence: a(n) = a(n-1) + 1, a(0) = 240.
  EBCDIC codes for decimal digits.
  Finite list; the displayed formula does not specify where the list stops.

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

Sequences:

• A058445 Apparent constant: a(n) = 2236081408416666.
  Numbers n such that n^2 contains only digits {0,5,6}, not ending with zero.
  A single listed term does not establish a constant sequence.
• A058446 Apparent constant: a(n) = 5000060065066660656065066555556.
  Squares composed of digits {0,5,6}, not ending with zero.
  A single listed term does not establish a constant sequence.
• A072288 Apparent constant: a(n) = 316912650057057350374175801344000001.
  Smallest prime factor greater than 16 of googolplex + n, or 1 if there is no such factor.
  A single listed term does not establish a constant sequence.
• A076337 Apparent constant: a(n) = 509203.
  Riesel numbers: odd n such that for all k ≥ 1 the numbers n*2^k - 1 are composite.
  A single listed term does not establish a constant sequence.
• A082710 Apparent constant: a(n) = 10907.
  Numbers k ≥ 3 such that (64*10^k-1+53)/9 is prime: the lengths of depression primes of the form 711...117.
  A single listed term does not establish a constant sequence.
• A115453 Apparent constant: a(n) = 1414213562373095048801688724209698078569671875376948073.
  Primes formed from the initial decimal digits of sqrt(2), with the decimal point omitted.
  A single listed term does not establish a constant sequence.
• A118329 Apparent constant: a(n) = 9159655941772190150546035149323841107741493742816721.
  Catalan-primes: primes formed from the concatenation of initial decimal digits of Catalan's constant.
  A single listed term does not establish a constant sequence.

Comments.

• The following observations are based on a comment sent to me by Max Alekseyev. The digit-pattern sequences listed above do not follow the displayed arithmetic progressions. For the bases b = 7, 8, 9, and 10 used above, the first values predicted by these progressions that do not belong to the sequences are as follows.
  • For the string 0,0 occurring in n but not in n-1, the first omitted value is b⁴+b²=b²*(b²+1).
  • For the string b-2,b-1 occurring in n but not in n-1, the first omitted value is (b²-b-1)*(b²+1).
  • For the string b-1,b-1 occurring in n but not in n+1, the first omitted value is b⁴-b²-1.

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Last revised October 17, 2026