A092464 Numbers congruent to 4 or 9 mod 13.

4, 9, 17, 22, 30, 35, 43, 48, 56, 61, 69, 74, 82, 87, 95, 100, 108, 113, 121, 126, 134, 139, 147, 152, 160, 165, 173, 178, 186, 191, 199, 204, 212, 217, 225, 230, 238, 243, 251, 256, 264, 269, 277, 282, 290, 295, 303, 308, 316, 321, 329, 334, 342, 347, 355, 360

Offset: 1

Author

Jun Mizuki (suzuki32(AT)sanken.osaka-u.ac.jp), Mar 25 2004

Keywords

• nonn
• easy

Comments

Numbers k such that k^2 is congruent to 3 (modulo 13).

Links

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

Crossrefs

A127547 is a subsequence.

Programs

Mathematica

Select[Range[400],MemberQ[{4,9},Mod[#,13]]&] (* or *) Select[Range[400], PowerMod[#,2,13]==3&] (* Harvey P. Dale, Mar 05 2012 *)

Formula

From R. J. Mathar, Apr 20 2009: (Start)

a(n) = a(n-2) + 13 = a(n-1) + a(n-2) - a(n-3) = 13*n/2 - 13/4 - 3*(-1)^n/4.

G.f.: x*(4+5*x+4*x^2)/((1+x)*(x-1)^2). (End)

a(n) = 13*(n-1) - a(n-1), (with a(1)=4). - Vincenzo Librandi, Nov 17 2010

Sum_{n>=1} (-1)^(n+1)/a(n) = tan(5*Pi/26)*Pi/13. - Amiram Eldar, Feb 27 2023

From Amiram Eldar, Nov 25 2024: (Start)

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

Product_{n>=1} (1 + (-1)^n/a(n)) = sin(3*Pi/13)*sec(5*Pi/26). (End)

Extensions

More terms from Ray Chandler, Mar 27 2004

Edited by N. J. A. Sloane, May 10 2007

Incorrect formula deleted by N. J. A. Sloane, Jun 16 2010