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

Author

Bruno Berselli, Feb 19 2016

Keywords

• nonn
• easy

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).

Crossrefs

Subsequence of A094784, A106389.

Cf. A140373.

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).

Sequences of the form 13*n+q: A008595 (q=0), A190991 (q=1), A153080 (q=2), A127547 (q=4), A154609 (q=5), A186113 (q=6), A269044 (q=7), this sequence (q=11).

Programs

Magma

[13*n+11: n in [0..60]];

Mathematica

13 Range[0,60] + 11
Range[11, 800, 13]
Table[13 n + 11, {n, 0, 60}] (* Bruno Berselli, Feb 22 2016 *)
LinearRecurrence[{2,-1},{11,24},60] (* Harvey P. Dale, Jun 14 2023 *)

Maxima

makelist(13*n+11, n, 0, 60);

PARI

vector(60, n, n--; 13*n+11)

Python

[13*n+11 for n in range(61)]

Sage

[13*n+11 for n in range(61)]

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.

E.g.f.: (11 + 13*x)*exp(x). - G. C. Greubel, May 31 2024