A114047 x such that x^2 - 13*y^2 = 1.

1, 649, 842401, 1093435849, 1419278889601, 1842222905266249, 2391203911756701601, 3103780835237293411849, 4028705132934095091878401, 5229256158767620191964752649, 6787570465375238075075157060001, 8810261234800900253827361899128649

Offset: 0

Author

Cino Hilliard, Feb 01 2006

Keywords

• nonn
• easy

Comments

A Pellian equation (Pell's equation). - Benoit Cloitre, Feb 03 2006

Numbers n such that 13*(n^2-1) is a square. - Vincenzo Librandi, Nov 13 2010

The corresponding values y of the solutions of this Pell equation are given in A075871(n). - Wolfdieter Lang, Jun 27 2013

Examples

			(649^2-1)/13 = 180^2.
		

Links

• Colin Barker, Table of n, a(n) for n = 0..321
• Tanya Khovanova, Recursive Sequences
• John Robertson, Home page.
• Index entries for linear recurrences with constant coefficients, signature (1298,-1).

Crossrefs

Cf. A202155, A075871, A132644.

Programs

Magma

I:=[1,649]; [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},{1,649},20] (* or *) With[{c=180Sqrt[13]}, Simplify[Table[1/2((649-c)^n+(649+c)^n),{n,0,20}]]] (* Harvey P. Dale, Aug 11 2011 *)

PARI

/* This sequence is computed with g(1e9,13) in the following program. */
g(n,k) = for(y=0,n,x=k*y^2+1;if(issquare(x),print1(floor(sqrt(x))",")))

PARI

a0=1;a1=649;for(n=2,30,a2=1298*a1-a0;a0=a1;a1=a2;print1(a2,",")) \\ Benoit Cloitre

PARI

Vec((1-649*x)/(1-1298*x+x^2) + O(x^100)) \\ Colin Barker, Jun 13 2015

Formula

a(0)=1, a(1)=649 then a(n)=1298*a(n-1)-a(n-2). - Benoit Cloitre, Feb 03 2006

G.f.: (1-649*x)/(1-1298*x+x^2). - Philippe Deléham, Nov 18 2008

a(n) = 2*A132644(n) + 1. - Hugo Pfoertner, Feb 11 2024

Extensions

More terms from Benoit Cloitre, Feb 03 2006