A155086 Numbers k such that k^2 == -1 (mod 13).

5, 8, 18, 21, 31, 34, 44, 47, 57, 60, 70, 73, 83, 86, 96, 99, 109, 112, 122, 125, 135, 138, 148, 151, 161, 164, 174, 177, 187, 190, 200, 203, 213, 216, 226, 229, 239, 242, 252, 255, 265, 268, 278, 281, 291, 294, 304, 307, 317, 320, 330, 333, 343, 346, 356, 359

Offset

1,1

Comments

Numbers k such that k == 5 or 8 mod 13. - Charles R Greathouse IV, Dec 28 2011

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

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

Formula

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

G.f.: x*(5+3*x+5*x^2)/((1+x)*(x-1)^2) .

a(n) = 13*(n-1/2)/2 -7*(-1)^n/4.

a(n) = a(n-2)+13. - M. F. Hasler, Jun 16 2010

Sum_{n>=1} (-1)^(n+1)/a(n) = tan(3*Pi/26)*Pi/13. - Amiram Eldar, Feb 27 2023

Mathematica

LinearRecurrence[{1, 1, -1}, {5, 8, 18}, 50] (* Vincenzo Librandi, Feb 26 2012 *)
Select[Range[1000], PowerMod[#, 2, 13] == 12 &] (* Vincenzo Librandi, Apr 24 2014 *)

Prog

(Magma) I:=[5, 8, 18]; [n le 3 select I[n] else Self(n-1)+Self(n-2)-Self(n-3): n in [1..40]]; // Vincenzo Librandi, Feb 26 2012

Crossrefs

Cf. A002144, A047221 (m=5), A155095 (m=17), A156619 (m=25), A155096 (m=29), A155097 (m=37), A155098 (m=41), A154609 (bisection).

Sequence in context: A359949 A291752 A237276 * A219049 A245534 A302393

Adjacent sequences: A155083 A155084 A155085 * A155087 A155088 A155089

Keyword

nonn,easy

Author

Vincenzo Librandi, Jan 20 2009

Extensions

Algebra simplified by R. J. Mathar, Aug 18 2009

Edited by N. J. A. Sloane, Jun 23 2010

Status

approved