A202155 -id:A202155 - cp's OEIS frontend

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

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

Offset: 0

Cino Hilliard, Feb 01 2006

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

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

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).

Cf. A202155, A075871, A132644.

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

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 *)

/* 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))",")))

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

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

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

More terms from Benoit Cloitre, Feb 03 2006

A202156 y-values in the solution to x^2 - 13*y^2 = -1.

Original entry on oeis.org

5, 6485, 8417525, 10925940965, 14181862955045, 18408047189707445, 23893631070377308565, 31013914721302556809925, 40256037414619648361974085, 52252305550261582271285552405, 67823452348202119168480285047605, 88034788895660800419105138706238885

Offset: 1

Bruno Berselli, Dec 15 2011

The corresponding values of x of this Pell equation are in A202155.

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

Bruno Berselli, Table of n, a(n) for n = 1..200
A. J. C. Cunningham, Binomial Factorisations, Vols. 1-9, Hodgson, London, 1923-1929. See Vol. 1, page xxxv.
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, A006191, A031396, A075871, A202155.

m:=13; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(5*x*(1-x)/(1-1298*x+x^2)));

LinearRecurrence[{1298, -1}, {5, 6485}, 12]

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

G.f.: 5*x*(1-x)/(1-1298*x+x^2).
a(n) = a(-n+1) = 5*(r^(2n-1)+1/r^(2n-1))/(r+1/r), where r=18+5*sqrt(13).
a(n) = A006191(6*n - 3). - Michael Somos, Feb 24 2023

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

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A202156 y-values in the solution to x^2 - 13*y^2 = -1.

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