A202155 x-values in the solution to x^2 - 13*y^2 = -1.

18, 23382, 30349818, 39394040382, 51133434066018, 66371158023650982, 86149711981264908618, 111822259780523827735182, 145145207045407947135357618, 188398366922679734857866452982, 244540935120431250437563520613018, 317413945387952840388222591889244382

Offset: 1

Author

Bruno Berselli, Dec 15 2011

Keywords

• nonn
• easy

Comments

The corresponding values of y of this Pell equation are in A202156.

References

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

Links

• Bruno Berselli, Table of n, a(n) for n = 1..200
• 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, A031396, A114047, A202156.

Programs

Magma

m:=13; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(18*x*(1+x)/(1-1298*x+x^2)));

Mathematica

LinearRecurrence[{1298, -1}, {18, 23382}, 12]

Maxima

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

Formula

G.f.: 18*x*(1+x)/(1-1298*x+x^2).

a(n) = -a(-n+1) = (r^(2n-1)-1/r^(2n-1))/2, where r=18+5*sqrt(13).