A097740 Chebyshev U(n,x) polynomial evaluated at x=201.

1, 402, 161603, 64964004, 26115368005, 10498312974006, 4220295700182407, 1696548373160353608, 682008225714761968009, 274165610188961150786010, 110213893287736667854008011, 44305710936059951516160434412

Offset: 0

Author

Wolfdieter Lang, Aug 31 2004

Keywords

• nonn
• easy

Comments

Used to form integer solutions of Pell equation a^2 - 101*b^2 =-1. See A097741 with A097742.

Links

• Indranil Ghosh, Table of n, a(n) for n = 0..383
• 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 (402, -1).

Programs

Mathematica

LinearRecurrence[{402, -1},{1, 402},12] (* Ray Chandler, Aug 11 2015 *)

Formula

a(n) = 2*201*a(n-1) - a(n-2), n>=1, a(0)=1, a(-1):=0.

a(n) = S(n, 2*201)= U(n, 201), Chebyshev's polynomials of the second kind. See A049310.

G.f.: 1/(1-402*x+x^2).

a(n)= sum((-1)^k*binomial(n-k, k)*402^(n-2*k), k=0..floor(n/2)), n>=0.

a(n) = ((201+20*sqrt(101))^(n+1) - (201-20*sqrt(101))^(n+1))/(40*sqrt(101)), n>=0.