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

Author

Bruno Berselli, Dec 15 2011

Keywords

• nonn
• easy

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).

Crossrefs

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

Programs

Magma

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

Mathematica

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

Maxima

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

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