A098245 Chebyshev polynomials S(n,227).

1, 227, 51528, 11696629, 2655083255, 602692202256, 136808474828857, 31054921093948283, 7049330279851431384, 1600166918605180975885, 363230841193096230094511, 82451800783914239050478112

Offset: 0

Author

Wolfdieter Lang, Sep 10 2004

Keywords

• nonn
• easy

Comments

Used for all positive integer solutions of Pell equation x^2 - 229*y^2 = -4. See A098246 with A098247.

Links

• Indranil Ghosh, Table of n, a(n) for n = 0..423
• Tanya Khovanova, Recursive Sequences
• Index entries for linear recurrences with constant coefficients, signature (227, -1).
• Index entries for sequences related to Chebyshev polynomials.

Programs

Mathematica

CoefficientList[Series[1/(1 - 227*x + x^2), {x, 0, 15}], x] (* Wesley Ivan Hurt, Feb 09 2017 *)
LinearRecurrence[{227,-1},{1,227},20] (* Harvey P. Dale, Jan 15 2019 *)

Formula

a(n) = S(n, 227) = U(n, 227/2) = S(2*n+1, sqrt(229))/sqrt(229) 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) = 227*a(n-1)-a(n-2), n >= 1; a(0)=1, a(1)=227; a(-1):=0.

a(n) = (ap^(n+1) - am^(n+1))/(ap-am) with ap := (227+15*sqrt(229))/2 and am := (227-15*sqrt(229))/2 = 1/ap.

G.f.: 1/(1-227*x+x^2).