A075871 Numbers k such that 13*k^2 + 1 is a square.

0, 180, 233640, 303264540, 393637139280, 510940703520900, 663200639532988920, 860833919173116097260, 1117361763886065161254560, 1450334708690193406192321620, 1882533334518107155172472208200, 2443526817869794397220462733921980

Offset: 1

Author

Gregory V. Richardson, Oct 16 2002

Keywords

• nonn
• easy

Comments

Limit_{n->infinity} a(n)/a(n-1) = 649 + 180*sqrt(13).

This sequence gives the values of y in solutions of the Diophantine equation x^2 - 13*y^2 = 1. The corresponding x values are in A114047. - Vincenzo Librandi, Aug 08 2010, edited by Jon E. Schoenfield, May 04 2014

Links

• Colin Barker, Table of n, a(n) for n = 1..322
• Tanya Khovanova, Recursive Sequences
• Index entries for linear recurrences with constant coefficients, signature (1298,-1).

Crossrefs

Cf. A114047.

Cf. A202156.

Programs

Magma

I:=[0,180]; [n le 2 select I[n] else 1298*Self(n-1)-Self(n-2): n in [1..20]]; // Vincenzo Librandi, Jun 14 2015

Mathematica

LinearRecurrence[{1298, -1}, {0, 180}, 50] (* Vincenzo Librandi, Jun 14 2015 *)

PARI

concat(0, Vec(180*x^2/(1-1298*x+x^2) + O(x^20))) \\ Colin Barker, Jun 13 2015

Formula

a(n) = 1298*a(n-1) - a(n-2), n>1. - Michael Somos, Oct 30 2002

a(n) = ((649 + 180*sqrt(13))^n - (649 - 180*sqrt(13))^n) / (2*sqrt(13)).

From Mohamed Bouhamida, Sep 20 2006: (Start)

a(n) = 1297*(a(n-1) + a(n-2)) - a(n-3).

a(n) = 1299*(a(n-1) - a(n-2)) + a(n-3). (End)

G.f.: 180*x^2/(1-1298*x+x^2). - Philippe Deléham, Nov 18 2008