A158640 a(n) = 52*n^2 - 1.

51, 207, 467, 831, 1299, 1871, 2547, 3327, 4211, 5199, 6291, 7487, 8787, 10191, 11699, 13311, 15027, 16847, 18771, 20799, 22931, 25167, 27507, 29951, 32499, 35151, 37907, 40767, 43731, 46799, 49971, 53247, 56627, 60111, 63699, 67391, 71187, 75087, 79091, 83199

Offset

1,1

Comments

The identity (52*n^2-1)^2 - (676*n^2-26)*(2*n)^2 = 1 can be written as a(n)^2 - A158639(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*(-51-54*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/(2*sqrt(13)))*Pi/(2*sqrt(13)))/2.

Sum_{n>=1} (-1)^(n+1)/a(n) = (cosec(Pi/(2*sqrt(13)))*Pi/(2*sqrt(13)) - 1)/2. (End)

Maple

A158640:=n->52*n^2 - 1; seq(A158640(n), n=1..50); # Wesley Ivan Hurt, Feb 22 2014

Mathematica

LinearRecurrence[{3, -3, 1}, {51, 207, 467}, 50] (* Vincenzo Librandi, Feb 17 2012 *)

Prog

(Magma) I:=[51, 207, 467]; [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(52*n^2 - 1", ")); \\ Vincenzo Librandi, Feb 17 2012

Crossrefs

Cf. A158639, A005843.

Sequence in context: A157365 A157916 A007264 * A107253 A030535 A201813

Adjacent sequences: A158637 A158638 A158639 * A158641 A158642 A158643

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