A097771 Chebyshev U(n,x) polynomial evaluated at x=339=2*13^2+1.

1, 678, 459683, 311664396, 211308000805, 143266512881394, 97134484425584327, 65857037174033292312, 44650974069510146603209, 30273294562090705363683390, 20525249062123428726430735211

Offset: 0

Author

Wolfdieter Lang, Aug 31 2004

Keywords

• nonn
• easy

Comments

Used to form integer solutions of Pell equation a^2 - 170*b^2 =-1. See A097772 with A097773.

Links

• Tanya Khovanova, Recursive Sequences
• Index entries for sequences related to Chebyshev polynomials.
• Index entries for linear recurrences with constant coefficients, signature (678, -1).

Programs

Mathematica

LinearRecurrence[{678, -1},{1, 678},11] (* Ray Chandler, Aug 12 2015 *)

Formula

a(n) = 2*339*a(n-1) - a(n-2), n>=1, a(0)=1, a(-1):=0.

a(n) = S(n, 2*339)= U(n, 339), Chebyshev's polynomials of the second kind. See A049310.

G.f.: 1/(1-2*339*x+x^2).

a(n)= sum((-1)^k*binomial(n-k, k)*678^(n-2*k), k=0..floor(n/2)), n>=0.

a(n) = ((339+26*sqrt(170))^(n+1) - (339-26*sqrt(170))^(n+1))/(52*sqrt(170)), n>=0.