A097765 Chebyshev U(n,x) polynomial evaluated at x=243=2*11^2+1.

1, 486, 236195, 114790284, 55787841829, 27112776338610, 13176753512722631, 6403875094406860056, 3112270119128221264585, 1512556874021221127728254, 735099528504194339854666859, 357256858296164427948240365220

Offset: 0

Author

Wolfdieter Lang, Aug 31 2004

Keywords

• nonn
• easy

Comments

Used to form integer solutions of Pell equation a^2 - 122*b^2 =-1. See A097766 with A097767.

Links

• Tanya Khovanova, Recursive Sequences
• Index entries for sequences related to Chebyshev polynomials.
• Index entries for linear recurrences with constant coefficients, signature (486, -1).

Programs

Mathematica

LinearRecurrence[{486, -1},{1, 486},12] (* Ray Chandler, Aug 12 2015 *)

Formula

a(n) = 2*243*a(n-1) - a(n-2), n>=1, a(0)=1, a(-1):=0.

a(n) = S(n, 2*243)= U(n, 243), Chebyshev's polynomials of the second kind. See A049310.

G.f.: 1/(1-486*x+x^2).

a(n)= sum((-1)^k*binomial(n-k, k)*486^(n-2*k), k=0..floor(n/2)), n>=0.

a(n) = ((243+22*sqrt(122))^(n+1) - (243-22*sqrt(122))^(n+1))/(44*sqrt(122)), n>=0.