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

1,2

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).

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(a(i), i=1..n-1) 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

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 *)

Prog

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

Crossrefs

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

Cf. A195045 (partial sums).

Sequence in context: A342491 A101557 A019292 * A181451 A022326 A307167

Adjacent sequences: A175883 A175884 A175885 * A175887 A175888 A175889

Keyword

nonn,easy

Author

Bruno Berselli, Oct 08 2010 - Nov 17 2010

Status

approved