A158693 a(n) = 66*n^2 - 1.

65, 263, 593, 1055, 1649, 2375, 3233, 4223, 5345, 6599, 7985, 9503, 11153, 12935, 14849, 16895, 19073, 21383, 23825, 26399, 29105, 31943, 34913, 38015, 41249, 44615, 48113, 51743, 55505, 59399, 63425, 67583, 71873, 76295, 80849, 85535, 90353, 95303, 100385, 105599

Offset

1,1

Comments

The identity (66*n^2 - 1)^2 - (1089*n^2 - 33)*(2*n)^2 = 1 can be written as a(n)^2 - A158692(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*(-65 - 68*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(66))*Pi/sqrt(66))/2.

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

Mathematica

LinearRecurrence[{3, -3, 1}, {65, 263, 593}, 50] (* Vincenzo Librandi, Feb 20 2012 *)

Prog

(Magma) I:=[65, 263, 593]; [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(66*n^2 - 1", ")); \\ Vincenzo Librandi, Feb 20 2012

Crossrefs

Cf. A005843, A158692.

Sequence in context: A152023 A369498 A165798 * A365874 A319617 A300162

Adjacent sequences: A158690 A158691 A158692 * A158694 A158695 A158696

Keyword

nonn,easy

Author

Vincenzo Librandi, Mar 24 2009

Extensions

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

Status

approved