A158656 a(n) = 54*n^2 - 1.

53, 215, 485, 863, 1349, 1943, 2645, 3455, 4373, 5399, 6533, 7775, 9125, 10583, 12149, 13823, 15605, 17495, 19493, 21599, 23813, 26135, 28565, 31103, 33749, 36503, 39365, 42335, 45413, 48599, 51893, 55295, 58805, 62423, 66149, 69983, 73925, 77975, 82133, 86399

Offset

1,1

Comments

The identity (54*n^2 - 1)^2 - (729*n^2 - 27)*(2*n)^2 = 1 can be written as a(n)^2 - A158655(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.: x*(-53 - 56*x + x^2)/(x-1)^3.

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

From Amiram Eldar, Mar 19 2023: (Start)

Sum_{n>=1} 1/a(n) = (1 - cot(Pi/(3*sqrt(6)))*Pi/(3*sqrt(6)))/2.

Sum_{n>=1} (-1)^(n+1)/a(n) = (cosec(Pi/(3*sqrt(6)))*Pi/(3*sqrt(6)) - 1)/2. (End)

Mathematica

LinearRecurrence[{3, -3, 1}, {53, 215, 485}, 50] (* Vincenzo Librandi, Feb 17 2012 *)
54*Range[40]^2-1 (* Harvey P. Dale, Sep 15 2021 *)

Prog

(Magma) I:=[53, 215, 485]; [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(54*n^2 - 1", ")); \\ Vincenzo Librandi, Feb 17 2012

Crossrefs

Cf. A005843, A158655.

Sequence in context: A330810 A005146 A158644 * A013536 A142000 A053652

Adjacent sequences: A158653 A158654 A158655 * A158657 A158658 A158659

Keyword

nonn,easy

Author

Vincenzo Librandi, Mar 23 2009

Extensions

Comment rephrased and redundant formula replaced by R. J. Mathar, Oct 19 2009

Status

approved