A158741 a(n) = 1369*n^2 + 37.

37, 1406, 5513, 12358, 21941, 34262, 49321, 67118, 87653, 110926, 136937, 165686, 197173, 231398, 268361, 308062, 350501, 395678, 443593, 494246, 547637, 603766, 662633, 724238, 788581, 855662, 925481, 998038, 1073333, 1151366, 1232137, 1315646, 1401893, 1490878

Offset

0,1

Comments

The identity (74*n^2 + 1)^2 - (1369*n^2 + 37)*(2*n)^2 = 1 can be written as A158742(n)^2 - a(n)*A005843(n)^2 = 1.

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..10000

Vincenzo Librandi, X^2-AY^2=1, Math Forum, 2007. [Wayback Machine link]

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Formula

G.f.: -37*(1 + 35*x + 38*x^2)/(x-1)^3.

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

From Amiram Eldar, Mar 23 2023: (Start)

Sum_{n>=0} 1/a(n) = (coth(Pi/sqrt(37))*Pi/sqrt(37) + 1)/74.

Sum_{n>=0} (-1)^n/a(n) = (cosech(Pi/sqrt(37))*Pi/sqrt(37) + 1)/74. (End)

Mathematica

LinearRecurrence[{3, -3, 1}, {37, 1406, 5513}, 50] (* Vincenzo Librandi, Feb 21 2012 *)

Prog

(Magma) I:=[37, 1406, 5513]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..40]]; // Vincenzo Librandi, Feb 21 2012
(PARI) for(n=0, 40, print1(1369*n^2 + 37", ")); \\ Vincenzo Librandi, Feb 21 2012

Crossrefs

Cf. A005843, A158742.

Sequence in context: A009981 A370334 A097315 * A094490 A231543 A279330

Adjacent sequences: A158738 A158739 A158740 * A158742 A158743 A158744

Keyword

nonn,easy

Author

Vincenzo Librandi, Mar 25 2009

Extensions

Comment rewritten, a(0) added and formula replaced by R. J. Mathar, Oct 22 2009

Status

approved