A175886 Numbers that are congruent to {1, 12} mod 13.

1, 12, 14, 25, 27, 38, 40, 51, 53, 64, 66, 77, 79, 90, 92, 103, 105, 116, 118, 129, 131, 142, 144, 155, 157, 168, 170, 181, 183, 194, 196, 207, 209, 220, 222, 233, 235, 246, 248, 259, 261, 272, 274, 285, 287, 298, 300, 311, 313, 324, 326, 337, 339, 350

Offset: 1

Author

Bruno Berselli, Oct 08 2010 - Nov 17 2010

Keywords

• nonn
• easy
• changed

Comments

Cf. property described by Gary Detlefs in A113801: more generally, these numbers are of the form (2*h*n+(h-4)*(-1)^n-h)/4 (h, n natural numbers), therefore ((2*h*n+(h-4)*(-1)^n-h)/4)^2-1 == 0 (mod h); in this case, a(n)^2-1 == 0 (mod 13).

Links

• Bruno Berselli, Table of n, a(n) for n = 1..10000
• Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Crossrefs

Cf. A000217, A091998, A113801, A005408, A047209, A007310, A047336, A047522, A056020, A090771, A175885, A175887.

Cf. A195045 (partial sums).

Programs

Haskell

a175886 n = a175886_list !! (n-1)
a175886_list = 1 : 12 : map (+ 13) a175886_list
-- Reinhard Zumkeller, Jan 07 2012

Magma

[n: n in [1..350] | n mod 13 in [1, 12]]; // Bruno Berselli, Feb 29 2012

Magma

[(26*n+9*(-1)^n-13)/4: n in [1..55]]; // Vincenzo Librandi, Aug 19 2013

Mathematica

Select[Range[1, 350], MemberQ[{1, 12}, Mod[#, 13]]&] (* Bruno Berselli, Feb 29 2012 *)
CoefficientList[Series[(1 + 11 x + x^2) / ((1 + x) (1 - x)^2), {x, 0, 55}], x] (* Vincenzo Librandi, Aug 19 2013 *)
LinearRecurrence[{1,1,-1},{1,12,14},60] (* Harvey P. Dale, Oct 23 2015 *)

PARI

a(n)=(26*n+9*(-1)^n-13)/4 \\ Charles R Greathouse IV, Sep 24 2015

Formula

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

a(n) = (26*n+9*(-1)^n-13)/4.

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

a(n) = a(n-2)+13.

a(n) = 13*A000217(n-1)+1 - 2*Sum_{i=1..n-1} a(i) for n>1.

Sum_{n>=1} (-1)^(n+1)/a(n) = (Pi/13)*cot(Pi/13). - Amiram Eldar, Dec 04 2021

E.g.f.: 1 + ((26*x - 13)*exp(x) + 9*exp(-x))/4. - David Lovler, Sep 04 2022

From Amiram Eldar, Nov 25 2024: (Start)

Product_{n>=1} (1 - (-1)^n/a(n)) = 2*cos(Pi/13).

Product_{n>=2} (1 + (-1)^n/a(n)) = (Pi/13)*cosec(Pi/13). (End)