A228137 Numbers that are congruent to {1, 4} mod 12.

1, 4, 13, 16, 25, 28, 37, 40, 49, 52, 61, 64, 73, 76, 85, 88, 97, 100, 109, 112, 121, 124, 133, 136, 145, 148, 157, 160, 169, 172, 181, 184, 193, 196, 205, 208, 217, 220, 229, 232, 241, 244, 253, 256, 265, 268, 277, 280, 289, 292, 301, 304, 313, 316, 325

Offset: 1

Author

Colin Barker, Aug 12 2013

Keywords

• nonn
• easy

Comments

The squares of the terms of A001651 are the squares of this sequence. - Bruno Berselli, Aug 12 2013

Links

• Colin Barker, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Crossrefs

Cf. A001651, A087445, A146512, A228138.

Programs

Mathematica

Select[Range[300], MemberQ[{1, 4}, Mod[#, 12]] &] (* Amiram Eldar, Dec 28 2021 *)

PARI

Vec(x*(8*x^2+3*x+1)/((x-1)^2*(x+1)) + O(x^99))

Formula

a(n) = -13/2 - 3*(-1)^n/2 + 6*n.

a(n) = a(n-1) + a(n-2) - a(n-3).

G.f.: x*(8*x^2+3*x+1) / ((x-1)^2*(x+1)).

Sum_{n>=1} (-1)^(n+1)/a(n) = (sqrt(3)+3)*Pi/36 + log(2)/4 - sqrt(3)*log(26-15*sqrt(3))/36. - Amiram Eldar, Dec 28 2021

E.g.f.: 8 + ((12*x - 13)*exp(x) - 3*exp(-x))/2. - David Lovler, Sep 04 2022