A158595 a(n) = 361*n^2 - 19.

342, 1425, 3230, 5757, 9006, 12977, 17670, 23085, 29222, 36081, 43662, 51965, 60990, 70737, 81206, 92397, 104310, 116945, 130302, 144381, 159182, 174705, 190950, 207917, 225606, 244017, 263150, 283005, 303582, 324881, 346902, 369645, 393110, 417297, 442206, 467837

Offset

1,1

Comments

The identity (38*n^2 - 1)^2 - (361*n^2 - 19)*(2*n)^2 = 1 can be written in as A158596(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

G.f.: 19*x*(-18 - 21*x + x^2)/(x-1)^3.

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

From Amiram Eldar, Mar 14 2023: (Start)

Sum_{n>=1} 1/a(n) = (1 - cot(Pi/sqrt(19))*Pi/sqrt(19))/38.

Sum_{n>=1} (-1)^(n+1)/a(n) = (cosec(Pi/sqrt(19))*Pi/sqrt(19) - 1)/38. (End)

Mathematica

LinearRecurrence[{3, -3, 1}, {342, 1425, 3230}, 50] (* Vincenzo Librandi, Feb 16 2012 *)

Prog

(Magma) I:=[342, 1425, 3230]; [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 16 2012
(PARI) for(n=1, 40, print1(361*n^2 - 19", ")); \\ Vincenzo Librandi, Feb 16 2012

Crossrefs

Cf. A005843, A158596.

Sequence in context: A043419 A252243 A252236 * A231267 A252235 A272380

Adjacent sequences: A158592 A158593 A158594 * A158596 A158597 A158598

Keyword

nonn,easy

Author

Vincenzo Librandi, Mar 22 2009

Extensions

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

Status

approved