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

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

Offset

1,1

Comments

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

References

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

Links

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).

Formula

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

Mathematica

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

Prog

(Magma) m:=13; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(5*x*(1-x)/(1-1298*x+x^2)));
(Maxima) makelist(expand(((18+5*sqrt(13))^(2*n-1)-(18-5*sqrt(13))^(2*n-1))/(2*sqrt(13))), n, 1, 12);

Crossrefs

Cf. A002313, A003654, A006191, A031396, A075871, A202155.

Sequence in context: A079812 A137694 A292334 * A117711 A203689 A116140

Adjacent sequences: A202153 A202154 A202155 * A202157 A202158 A202159

Keyword

nonn,easy

Author

Bruno Berselli, Dec 15 2011

Status

approved