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

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

Offset

0,2

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

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).

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

Example

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

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 *)

Prog

(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
(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

Crossrefs

Cf. A202155, A075871, A132644.

Sequence in context: A154358 A252538 A185666 * A288142 A157915 A158639

Adjacent sequences: A114044 A114045 A114046 * A114048 A114049 A114050

Keyword

nonn,easy

Author

Cino Hilliard, Feb 01 2006

Extensions

More terms from Benoit Cloitre, Feb 03 2006

Status

approved