A158627 a(n) = 484*n^2 - 22.

462, 1914, 4334, 7722, 12078, 17402, 23694, 30954, 39182, 48378, 58542, 69674, 81774, 94842, 108878, 123882, 139854, 156794, 174702, 193578, 213422, 234234, 256014, 278762, 302478, 327162, 352814, 379434, 407022, 435578, 465102, 495594, 527054, 559482, 592878

Offset

1,1

Comments

The identity (44*n^2-1)^2 - (484*n^2-22)*(2*n)^2 = 1 can be written as A158628(n)^2 - a(n)*A005843(n)^2 = 1.

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..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

From - Vincenzo Librandi, Feb 17 2012: (Start)

G.f.: 22*x*(-21-24*x+x^2)/(x-1)^3.

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

From Amiram Eldar, Mar 16 2023: (Start)

Sum_{n>=1} 1/a(n) = (1 - cot(Pi/sqrt(22))*Pi/sqrt(22))/44.

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

Mathematica

LinearRecurrence[{3, -3, 1}, {462, 1914, 4334}, 50] (* Vincenzo Librandi, Feb 17 2012 *)

Prog

(Magma) I:=[462, 1914, 4334]; [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=1, 40, print1(484*n^2-22", ")); \\ Vincenzo Librandi, Feb 17 2012

Crossrefs

Cf. A005843, A158628.

Sequence in context: A222342 A246476 A202642 * A267567 A267557 A154056

Adjacent sequences: A158624 A158625 A158626 * A158628 A158629 A158630

Keyword

nonn,easy

Author

Vincenzo Librandi, Mar 23 2009

Extensions

Edited by R. J. Mathar, Jul 26 2009

Status

approved