A158689 a(n) = 66*n^2 + 1.

1, 67, 265, 595, 1057, 1651, 2377, 3235, 4225, 5347, 6601, 7987, 9505, 11155, 12937, 14851, 16897, 19075, 21385, 23827, 26401, 29107, 31945, 34915, 38017, 41251, 44617, 48115, 51745, 55507, 59401, 63427, 67585, 71875, 76297, 80851, 85537, 90355, 95305, 100387

Offset

0,2

Comments

The identity (66*n^2 + 1)^2 - (1089*n^2 + 33)*(2*n)^2 = 1 can be written as a(n)^2 - A158688(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.: -(1 + 64*x + 67*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>=0} 1/a(n) = (coth(Pi/sqrt(66))*Pi/sqrt(66) + 1)/2.

Sum_{n>=0} (-1)^n/a(n) = (cosech(Pi/sqrt(66))*Pi/sqrt(66) + 1)/2. (End)

Mathematica

LinearRecurrence[{3, -3, 1}, {1, 67, 265}, 50] (* Vincenzo Librandi, Feb 20 2012 *)

Prog

(Magma) I:=[1, 67, 265]; [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 20 2012
(PARI) for(n=0, 40, print1(66*n^2 + 1", ")); \\ Vincenzo Librandi, Feb 20 2012

Crossrefs

Cf. A005843, A158688.

Sequence in context: A142273 A141985 A142429 * A142486 A158730 A140731

Adjacent sequences: A158686 A158687 A158688 * A158690 A158691 A158692

Keyword

nonn,easy

Author

Vincenzo Librandi, Mar 24 2009

Extensions

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

Status

approved