A158596 a(n) = 38*n^2 - 1.

37, 151, 341, 607, 949, 1367, 1861, 2431, 3077, 3799, 4597, 5471, 6421, 7447, 8549, 9727, 10981, 12311, 13717, 15199, 16757, 18391, 20101, 21887, 23749, 25687, 27701, 29791, 31957, 34199, 36517, 38911, 41381, 43927, 46549, 49247, 52021, 54871, 57797, 60799, 63877

Offset

1,1

Comments

The identity (38*n^2 - 1)^2 - (361*n^2 - 19)*(2*n)^2 = 1 can be written as a(n)^2 - A158595(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*(-37 - 40*x + x^2)/(x-1)^3.

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

From Amiram Eldar, Mar 14 2023: (Start)

Sum_{n>=1} 1/a(n) = (1 - cot(Pi/sqrt(38))*Pi/sqrt(38))/2.

Sum_{n>=1} (-1)^(n+1)/a(n) = (cosec(Pi/sqrt(38))*Pi/sqrt(38) - 1)/2. (End)

Mathematica

LinearRecurrence[{3, -3, 1}, {37, 151, 341}, 50] (* Vincenzo Librandi, Feb 16 2012 *)

Prog

(Magma) I:=[37, 151, 341]; [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 16 2012
(PARI) for(n=1, 40, print1(38*n^2 - 1", ")); \\ Vincenzo Librandi, Feb 16 2012

Crossrefs

Cf. A005843, A158595.

Sequence in context: A142656 A145898 A096700 * A142688 A178835 A250942

Adjacent sequences: A158593 A158594 A158595 * A158597 A158598 A158599

Keyword

nonn,easy

Author

Vincenzo Librandi, Mar 22 2009

Extensions

Comment rewritten, formula replaced by R. J. Mathar, Oct 28 2009

Status

approved