A097768 Chebyshev U(n,x) polynomial evaluated at x=289=2*12^2+1.

1, 578, 334083, 193099396, 111611116805, 64511032413894, 37287265124113927, 21551974730705435912, 12457004107082617843209, 7200126821919022407938890, 4161660846065087869170835211, 2405432768898798869358334813068

Offset: 0

Author

Wolfdieter Lang, Aug 31 2004

Keywords

• nonn
• easy

Comments

Used to form integer solutions of Pell equation a^2 - 145*b^2 =-1. See A097769 with A097770.

Links

• Tanya Khovanova, Recursive Sequences
• Index entries for sequences related to Chebyshev polynomials.
• Index entries for linear recurrences with constant coefficients, signature (578,-1).

Crossrefs

Cf. A049310, A097769, A097770.

Programs

Mathematica

LinearRecurrence[{578, -1},{1, 578},12] (* Ray Chandler, Aug 12 2015 *)

PARI

a(n) = polchebyshev(n, 2, 289); \\ Michel Marcus, Jun 20 2020

Formula

a(n) = 2*289*a(n-1) - a(n-2), n>=1, a(0)=1, a(-1):=0.

a(n) = S(n, 2*289)= U(n, 289), Chebyshev's polynomials of the second kind. See A049310.

G.f.: 1/(1-2*289*x+x^2).

a(n)= sum((-1)^k*binomial(n-k, k)*578^(n-2*k), k=0..floor(n/2)), n>=0.

a(n) = ((289+24*sqrt(145))^(n+1) - (289-24*sqrt(145))^(n+1))/(48*sqrt(145)), n>=0.