A158692 a(n) = 1089*n^2 - 33.

1056, 4323, 9768, 17391, 27192, 39171, 53328, 69663, 88176, 108867, 131736, 156783, 184008, 213411, 244992, 278751, 314688, 352803, 393096, 435567, 480216, 527043, 576048, 627231, 680592, 736131, 793848, 853743, 915816, 980067, 1046496, 1115103, 1185888, 1258851

Offset

1,1

Comments

The identity (66*n^2 - 1)^2 - (1089*n^2 - 33)*(2*n)^2 = 1 can be written as A158693(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.: 33*x*(-32 - 35*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(33))*Pi/sqrt(33))/66.

Sum_{n>=1} (-1)^(n+1)/a(n) = (cosec(Pi/sqrt(33))*Pi/sqrt(33) - 1)/66. (End)

Mathematica

LinearRecurrence[{3, -3, 1}, {1056, 4323, 9768}, 50] (* Vincenzo Librandi, Feb 20 2012 *)
1089*Range[40]^2-33 (* Harvey P. Dale, Nov 29 2017 *)

Prog

(Magma) I:=[1056, 4323, 9768]; [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=1, 40, print1(1089*n^2 - 33", ")); \\ Vincenzo Librandi, Feb 20 2012

Crossrefs

Cf. A005843, A158693.

Sequence in context: A339249 A359624 A225714 * A236905 A281079 A023072

Adjacent sequences: A158689 A158690 A158691 * A158693 A158694 A158695

Keyword

nonn,easy

Author

Vincenzo Librandi, Mar 24 2009

Extensions

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

Status

approved