A158679 a(n) = 961*n^2 - 31.

930, 3813, 8618, 15345, 23994, 34565, 47058, 61473, 77810, 96069, 116250, 138353, 162378, 188325, 216194, 245985, 277698, 311333, 346890, 384369, 423770, 465093, 508338, 553505, 600594, 649605, 700538, 753393, 808170, 864869, 923490, 984033, 1046498, 1110885, 1177194

Offset

1,1

Comments

The identity (62*n^2 - 1)^2 - (961*n^2 - 31)*(2*n)^2 = 1 can be written as A158680(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.: 31*x*(-30 - 33*x + x^2)/(x-1)^3.

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

From Amiram Eldar, Mar 21 2023: (Start)

Sum_{n>=1} 1/a(n) = (1 - cot(Pi/sqrt(31))*Pi/sqrt(31))/62.

Sum_{n>=1} (-1)^(n+1)/a(n) = (cosec(Pi/sqrt(31))*Pi/sqrt(31) - 1)/62. (End)

Mathematica

LinearRecurrence[{3, -3, 1}, {930, 3813, 8618}, 50] (* Vincenzo Librandi, Feb 19 2012 *)

Prog

(Magma) I:=[930, 3813, 8618]; [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 19 2012
(PARI) for(n=1, 40, print1(961*n^2 - 31", ")); \\ Vincenzo Librandi, Feb 19 2012

Crossrefs

Cf. A005843, A158680.

Sequence in context: A068651 A175726 A115958 * A250384 A209810 A035856

Adjacent sequences: A158676 A158677 A158678 * A158680 A158681 A158682

Keyword

nonn,easy

Author

Vincenzo Librandi, Mar 24 2009

Extensions

Comment rewritten and formula replaced by R. J. Mathar, Oct 22 2009

Status

approved