A098260 Chebyshev polynomials S(n,627).

1, 627, 393128, 246490629, 154549231255, 96902121506256, 60757475635191257, 38094840321143411883, 23885404123881284059384, 14976110290833243961821885, 9389997266948320082778262511

Offset: 0

Author

Wolfdieter Lang, Sep 10 2004

Keywords

• nonn
• easy

Comments

Used for all positive integer solutions of Pell equation x^2 - 629*y^2 = -4. See A098261 with A098262.

Links

• Tanya Khovanova, Recursive Sequences
• Index entries for sequences related to Chebyshev polynomials.
• Index entries for linear recurrences with constant coefficients, signature (627, -1).

Programs

Mathematica

LinearRecurrence[{627,-1},{1,627},20] (* Harvey P. Dale, Aug 28 2012 *)

Formula

a(n)= S(n, 627)=U(n, 627/2)= S(2*n+1, sqrt(629))/sqrt(629) with S(n, x)=U(n, x/2) Chebyshev's polynomials of the second kind, A049310. S(-1, x)= 0 = U(-1, x).

a(n)=627*a(n-1)-a(n-2), n >= 1; a(0)=1, a(1)=627; a(-1):=0.

a(n)=(ap^(n+1) - am^(n+1))/(ap-am) with ap := (627+25*sqrt(629))/2 and am := (627-25*sqrt(629))/2 = 1/ap.

G.f.: 1/(1-627*x+x^2).