A097774 Chebyshev U(n,x) polynomial evaluated at x=393=2*14^2+1.

1, 786, 617795, 485586084, 381670044229, 299992169177910, 235793463303793031, 185333362164612144456, 145671786867921841749385, 114497839144824403002872154, 89995155896045112838415763659

Offset: 0

Author

Wolfdieter Lang, Aug 31 2004

Keywords

• nonn
• easy

Comments

Used to form integer solutions of Pell equation a^2 - 197*b^2 =-1. See A097775 with A097776.

Links

• Tanya Khovanova, Recursive Sequences
• Index entries for sequences related to Chebyshev polynomials.
• Index entries for linear recurrences with constant coefficients, signature (786, -1).

Programs

Mathematica

LinearRecurrence[{786,-1},{1,786},30] (* or *) CoefficientList[ Series[ 1/(1-786x+x^2), {x,0,30}],x] (* Harvey P. Dale, Jun 15 2011 *)

Formula

a(n) = 2*393*a(n-1) - a(n-2), n>=1, a(0)=1, a(-1):=0.

a(n) = S(n, 2*393)= U(n, 393), Chebyshev's polynomials of the second kind. See A049310.

G.f.: 1/(1-2*393*x+x^2).

a(n)= sum((-1)^k*binomial(n-k, k)*786^(n-2*k), k=0..floor(n/2)), n>=0.

a(n) = ((393+28*sqrt(197))^(n+1) - (393-28*sqrt(197))^(n+1))/(56*sqrt(197)), n>=0.