A098254 Chebyshev polynomials S(n,443).

1, 443, 196248, 86937421, 38513081255, 17061208058544, 7558076656853737, 3348210897778146947, 1483249869639062243784, 657076344039206795849365, 291083337159498971499024911, 128949261285314005167272186208

Offset: 0

Author

Wolfdieter Lang, Sep 10 2004

Keywords

• nonn
• easy

Comments

Used for all positive integer solutions of Pell equation x^2 - 445*y^2 = -4. See A098255 with A098256.

Links

• Indranil Ghosh, Table of n, a(n) for n = 0..377
• Tanya Khovanova, Recursive Sequences
• Index entries for linear recurrences with constant coefficients, signature (443, -1).
• Index entries for sequences related to Chebyshev polynomials.

Formula

G.f.: 1/(1 - 443*x + x^2).

a(n) = S(n, 443)=U(n, 443/2)= S(2*n+1, sqrt(445))/sqrt(445) 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) = 443*a(n-1)-a(n-2) for n >= 1, a(0)=1, a(1)=443, and a(-1):=0.

a(n) = (ap^(n+1) - am^(n+1))/(ap - am) with ap:=(443 + 21*sqrt(445))/2 and am:=(443 - 21*sqrt(445))/2 = 1/ap.