A158592 a(n) = 361*n^2 + 19.

19, 380, 1463, 3268, 5795, 9044, 13015, 17708, 23123, 29260, 36119, 43700, 52003, 61028, 70775, 81244, 92435, 104348, 116983, 130340, 144419, 159220, 174743, 190988, 207955, 225644, 244055, 263188, 283043, 303620, 324919, 346940, 369683, 393148, 417335, 442244

Offset

0,1

Comments

The identity (38*n^2 + 1)^2 - (361*n^2 + 19)*(2*n)^2 = 1 can be written as A158593(n)^2 - a(n)*A005843(n)^2 = 1.

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..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.: -19*(1 + 17*x + 20*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>=0} 1/a(n) = (coth(Pi/sqrt(19))*Pi/sqrt(19) + 1)/38.

Sum_{n>=0} (-1)^n/a(n) = (cosech(Pi/sqrt(19))*Pi/sqrt(19) + 1)/38. (End)

Mathematica

LinearRecurrence[{3, -3, 1}, {19, 380, 1463}, 40] (* Vincenzo Librandi, Feb 16 2012 *)

Prog

(Magma) I:=[19, 380, 1463]; [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=0, 40, print1(361*n^2 + 19", ")); \\ Vincenzo Librandi, Feb 16 2012

Crossrefs

Cf. A158593, A005843.

Sequence in context: A263371 A023283 A075839 * A072359 A222835 A094737

Adjacent sequences: A158589 A158590 A158591 * A158593 A158594 A158595

Keyword

nonn,easy

Author

Vincenzo Librandi, Mar 22 2009

Extensions

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

Status

approved