A202156 - OEIS

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

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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