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

Author

Vincenzo Librandi, Jan 20 2009

Keywords

• nonn
• easy

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).

Crossrefs

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

Programs

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

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 *)

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

From Amiram Eldar, Nov 25 2024: (Start)

Product_{n>=1} (1 - (-1)^n/a(n)) = cos(Pi/26)*sec(3*Pi/26) = 1/(2*cos(Pi/13)-1).

Product_{n>=1} (1 + (-1)^n/a(n)) = cosec(5*Pi/26)/2. (End)

Extensions

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

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