A097734 - OEIS

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A097734

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

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

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OFFSET

0,2

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).

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.

MATHEMATICA

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

CROSSREFS