A269100 a(n) = 13*n + 11.

11, 24, 37, 50, 63, 76, 89, 102, 115, 128, 141, 154, 167, 180, 193, 206, 219, 232, 245, 258, 271, 284, 297, 310, 323, 336, 349, 362, 375, 388, 401, 414, 427, 440, 453, 466, 479, 492, 505, 518, 531, 544, 557, 570, 583, 596, 609, 622, 635, 648, 661, 674, 687, 700, 713, 726, 739

Offset

0,1

Comments

Any square mod 13 is one of 0, 1, 3, 4, 9, 10 or 12 (A010376) but not 11, and for this reason there are no squares in the sequence. Likewise, any cube mod 13 is one of 0, 1, 5, 8 or 12, therefore no a(k) is a cube.

Sequences of the type 13*n + k, for k = 0..12, without squares and cubes:

k = 2: A153080,

k = 6: A186113,

k = 7: A269044,

k = 11: this case.

The sum of the sixth powers of any two terms of the sequence is also a term of the sequence. Example: a(3)^6 + a(8)^6 = a(179129674278) = 2328685765625.

The primes of the sequence are listed in A140373.

Links

Bruno Berselli, Table of n, a(n) for n = 0..1000

Tanya Khovanova, Recursive Sequences.

Index entries for linear recurrences with constant coefficients, signature (2,-1).

Formula

G.f.: (11 + 2*x)/(1 - x)^2.

a(n) = -A153080(-n-1).

Sum_{i = h..h+13*k} a(i) = a(h*(13*k + 1) + k*(169*k + 35)/2).

Sum_{i >= 0} 1/a(i)^2 = .012486605016510955990... = polygamma(1, 11/13)/13^2.

Mathematica

13 Range[0, 60] + 11 (* or *) Range[11, 800, 13] (* or *) Table[13 n + 11, {n, 0, 60}]
LinearRecurrence[{2, -1}, {11, 24}, 60] (* Harvey P. Dale, Jun 14 2023 *)

Prog

(PARI) vector(60, n, n--; 13*n+11)
(Sage) [13*n+11 for n in range(60)]
(Python) [13*n+11 for n in range(60)]
(Maxima) makelist(13*n+11, n, 0, 60);
(Magma) [13*n+11: n in [0..60]];

Crossrefs

Subsequence of A094784, A106389.

Cf. A140373, A153080, A186113, A269044.

Cf. similar sequences of the type k*n+k-2: A023443 (k=1), A005843 (k=2), A016777 (k=3), A016825 (k=4), A016885 (k=5), A016957 (k=6), A017041 (k=7), A017137 (k=8), A017245 (k=9), A017365 (k=10), A017497 (k=11), A017641 (k=12).

Sequence in context: A112134 A063307 A213991 * A213062 A157756 A061043

Adjacent sequences: A269097 A269098 A269099 * A269101 A269102 A269103

Keyword

nonn,easy

Author

Bruno Berselli, Feb 19 2016

Status

approved