A158629 a(n) = 484*n^2 + 22.

22, 506, 1958, 4378, 7766, 12122, 17446, 23738, 30998, 39226, 48422, 58586, 69718, 81818, 94886, 108922, 123926, 139898, 156838, 174746, 193622, 213466, 234278, 256058, 278806, 302522, 327206, 352858, 379478, 407066, 435622, 465146, 495638, 527098, 559526, 592922

Offset

0,1

Comments

The identity (44*n^2 + 1)^2 - (484*n^2 + 22)*(2*n)^2 = 1 can be written as A158630(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.: -22*(1 + 20*x + 23*x^2)/(x-1)^3.

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

From Amiram Eldar, Mar 16 2023: (Start)

Sum_{n>=0} 1/a(n) = (coth(Pi/sqrt(22))*Pi/sqrt(22) + 1)/44.

Sum_{n>=0} (-1)^n/a(n) = (cosech(Pi/sqrt(22))*Pi/sqrt(22) + 1)/44. (End)

Mathematica

LinearRecurrence[{3, -3, 1}, {22, 506, 1958}, 50] (* Vincenzo Librandi, Feb 17 2012 *)

Prog

(Magma) I:=[22, 506, 1958]; [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 17 2012
(PARI) for(n=0, 40, print1(484*n^2 + 22", ")); \\ Vincenzo Librandi, Feb 17 2012

Crossrefs

Cf. A005843, A158630.

Sequence in context: A230350 A180780 A121904 * A253777 A266884 A182610

Adjacent sequences: A158626 A158627 A158628 * A158630 A158631 A158632

Keyword

nonn,easy

Author

Vincenzo Librandi, Mar 23 2009

Extensions

Comment rewritten, formula replaced by R. J. Mathar, Oct 28 2009

Status

approved