A097734 Chebyshev U(n,x) polynomial evaluated at x=129 = 3*43.

1, 258, 66563, 17172996, 4430566405, 1143068959494, 294907360983047, 76084956064666632, 19629623757323008009, 5064366844433271399690, 1306587016240026698112011, 337094385823082454841499148

Offset: 0

Author

Wolfdieter Lang, Aug 31 2004

Keywords

• nonn
• easy

Comments

Used to form integer solutions of Pell equation a^2 - 65*b^2 =-1. See A097735 with A097736.

Links

• Indranil Ghosh, Table of n, a(n) for n = 0..413
• R. Flórez, R. A. Higuita, A. Mukherjee, Alternating Sums in the Hosoya Polynomial Triangle, Article 14.9.5 Journal of Integer Sequences, Vol. 17 (2014).
• Tanya Khovanova, Recursive Sequences
• Index entries for sequences related to Chebyshev polynomials.
• Index entries for linear recurrences with constant coefficients, signature (258, -1).

Programs

Mathematica

LinearRecurrence[{258, -1},{1, 258},12] (* Ray Chandler, Aug 11 2015 *)

Formula

a(n) = 2*129*a(n-1) - a(n-2), n>=1, a(0)=1, a(-1):=0.

a(n) = S(n, 2*129)= U(n, 129), Chebyshev's polynomials of the second kind. See A049310.

G.f.: 1/(1-258*x+x^2).

a(n)= sum((-1)^k*binomial(n-k, k)*258^(n-2*k), k=0..floor(n/2)), n>=0.

a(n) = ((129+16*sqrt(65))^(n+1) - (129-16*sqrt(65))^(n+1))/(32*sqrt(65)), n>=0.