A158639 a(n) = 676*n^2 - 26.

650, 2678, 6058, 10790, 16874, 24310, 33098, 43238, 54730, 67574, 81770, 97318, 114218, 132470, 152074, 173030, 195338, 218998, 244010, 270374, 298090, 327158, 357578, 389350, 422474, 456950, 492778, 529958, 568490, 608374, 649610, 692198, 736138, 781430, 828074

Offset

1,1

Comments

The identity (52*n^2 - 1)^2 - (676*n^2 - 26)*(2*n)^2 = 1 can be written as A158640(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.: 26*x*(-25 - 28*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/sqrt(26))*Pi/sqrt(26))/52.

Sum_{n>=1} (-1)^(n+1)/a(n) = (cosec(Pi/sqrt(26))*Pi/sqrt(26) - 1)/52. (End)

Mathematica

LinearRecurrence[{3, -3, 1}, {650, 2678, 6058}, 50] (* Vincenzo Librandi, Feb 17 2012 *)

Prog

(Magma) I:=[650, 2678, 6058]; [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(676*n^2 - 26", ")); \\ Vincenzo Librandi, Feb 17 2012

Crossrefs

Cf. A158640, A005843.

Sequence in context: A114047 A288142 A157915 * A162025 A251861 A035851

Adjacent sequences: A158636 A158637 A158638 * A158640 A158641 A158642

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