A113541 Numbers k such that 18*k^2+1 is multiple of 19.

1, 18, 20, 37, 39, 56, 58, 75, 77, 94, 96, 113, 115, 132, 134, 151, 153, 170, 172, 189, 191, 208, 210, 227, 229, 246, 248, 265, 267, 284, 286, 303, 305, 322, 324, 341, 343, 360, 362, 379, 381, 398, 400, 417, 419, 436, 438, 455, 457, 474, 476, 493, 495, 512

Offset: 1

Author

Zak Seidov, Jan 13 2006

Keywords

• nonn
• easy

Comments

Numbers congruent to {1, 18} mod 19. - Amiram Eldar, Feb 27 2023

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Programs

Mathematica

s={};Do[If[IntegerQ[(18n^2+1)/19], AppendTo[s, n]], {n, 200}];s

PARI

a(n)=([0,1,0; 0,0,1; -1,1,1]^(n-1)*[1;18;20])[1,1] \\ Charles R Greathouse IV, Jul 08 2024

Formula

a(1)=1, a(2)=18 then add alternatively 2 and 17.

a(n) = 19*(n-1) - a(n-1), (with a(1)=1). - Vincenzo Librandi, Nov 13 2010

Sum_{n>=1} (-1)^(n+1)/a(n) = cot(Pi/19)*Pi/19. - Amiram Eldar, Feb 27 2023

Extensions

a(22)-a(56) from Vincenzo Librandi, Oct 23 2009