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A114047 x such that x^2 - 13*y^2 = 1. 4
1, 649, 842401, 1093435849, 1419278889601, 1842222905266249, 2391203911756701601, 3103780835237293411849, 4028705132934095091878401, 5229256158767620191964752649, 6787570465375238075075157060001, 8810261234800900253827361899128649 (list; graph; refs; listen; history; text; internal format)
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

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 23015

CROSSREFS

Cf. A202155, A075871.

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

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Last modified June 24 20:12 EDT 2017. Contains 288707 sequences.