A097737 Chebyshev U(n,x) polynomial evaluated at x=163.

1, 326, 106275, 34645324, 11294269349, 3681897162450, 1200287180689351, 391289939007565976, 127559319829285818825, 41583946974408169370974, 13556239154337233929118699, 4419292380366963852723324900

Offset: 0

Author

Wolfdieter Lang, Aug 31 2004

Keywords

• nonn
• easy

Comments

Used to form integer solutions of Pell equation a^2 - 82*b^2 =-1. See A097738 with A097739.

Links

• Indranil Ghosh, Table of n, a(n) for n = 0..397
• 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 (326, -1).

Programs

Mathematica

LinearRecurrence[{326, -1},{1, 326},12] (* Ray Chandler, Aug 11 2015 *)

Formula

a(n) = 2*163*a(n-1) - a(n-2), n>=1, a(0)=1, a(-1):=0.

a(n) = S(n, 2*163)= U(n, 163), Chebyshev's polynomials of the second kind. See A049310.

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

a(n)= sum((-1)^k*binomial(n-k, k)*326^(n-2*k), k=0..floor(n/2)), n>=0.

G.f.: 1/(1-326*x+x^2).

a(n) = ((163+18*sqrt(82))^(n+1) - (163-18*sqrt(82))^(n+1))/(36*sqrt(82)), n>=0.