A098251 Chebyshev polynomials S(n,363).

1, 363, 131768, 47831421, 17362674055, 6302602850544, 2287827472073417, 830475069759799827, 301460162495335263784, 109429208510736940953765, 39722501229235014230952911, 14419158517003799428894952928

Offset: 0

Author

Wolfdieter Lang, Sep 10 2004

Keywords

• nonn
• easy

Comments

Used for all positive integer solutions of Pell equation x^2 - 365*y^2 = -4. See A098252 with A098253.

Links

• Indranil Ghosh, Table of n, a(n) for n = 0..389
• Tanya Khovanova, Recursive Sequences
• Index entries for linear recurrences with constant coefficients, signature (363,-1).
• Index entries for sequences related to Chebyshev polynomials.

Formula

a(n)= S(n, 363)=U(n, 363/2)= S(2*n+1, sqrt(365))/sqrt(365) 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)=363*a(n-1)-a(n-2), n >= 1; a(0)=1, a(1)=363; a(-1):=0.

a(n)=(ap^(n+1) - am^(n+1))/(ap-am) with ap := (363+19*sqrt(365))/2 and am := (363-19*sqrt(365))/2 = 1/ap.

G.f.: 1/(1-363*x+x^2).