A202156 -id:A202156 - cp's OEIS frontend

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

Gregory V. Richardson, Oct 16 2002

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

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).

Cf. A114047.

Cf. A202156.

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

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

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

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

A202155 x-values in the solution to x^2 - 13*y^2 = -1.

Original entry on oeis.org

18, 23382, 30349818, 39394040382, 51133434066018, 66371158023650982, 86149711981264908618, 111822259780523827735182, 145145207045407947135357618, 188398366922679734857866452982, 244540935120431250437563520613018, 317413945387952840388222591889244382

Offset: 1

Bruno Berselli, Dec 15 2011

The corresponding values of y of this Pell equation are in A202156.

A. H. Beiler, Recreations in the Theory of Numbers: The Queen of Mathematics Entertains, Dover (New York), 1966, p. 264.

Bruno Berselli, Table of n, a(n) for n = 1..200
Tanya Khovanova, Recursive Sequences.
A. M. S. Ramasamy, Polynomial solutions for the Pell's equation, Indian Journal of Pure and Applied Mathematics 25 (1994), p. 579 (Theorem 4, case t=1).
J. P. Robertson, Solving the generalized Pell equation x^2-D*y^2=N, pp. 9, 24.
Index entries for linear recurrences with constant coefficients, signature (1298,-1).

Cf. A002313, A003654, A031396, A114047, A202156.

m:=13; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(18*x*(1+x)/(1-1298*x+x^2)));

LinearRecurrence[{1298, -1}, {18, 23382}, 12]

makelist(expand(((18+5*sqrt(13))^(2*n-1)+(18-5*sqrt(13))^(2*n-1))/2), n, 1, 12);

G.f.: 18*x*(1+x)/(1-1298*x+x^2).

a(n) = -a(-n+1) = (r^(2n-1)-1/r^(2n-1))/2, where r=18+5*sqrt(13).

cp's OEIS Frontend

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

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