A098263 Chebyshev polynomials S(n,731).

1, 731, 534360, 390616429, 285540075239, 208729404383280, 152580909064102441, 111536435796454501091, 81532981986299176195080, 59600498295548901344102389, 43567882721064260583362651279

Offset: 0

Author

Wolfdieter Lang, Sep 10 2004

Keywords

• nonn
• easy

Comments

Used for all positive integer solutions of Pell equation x^2 - 733*y^2 = -4. See A098291 with A098292.

Links

• Tanya Khovanova, Recursive Sequences
• Index entries for linear recurrences with constant coefficients, signature (731, -1).
• Index entries for sequences related to Chebyshev polynomials.

Programs

Mathematica

LinearRecurrence[{731,-1},{1,731},20] (* Harvey P. Dale, Jun 21 2020 *)

Formula

a(n)= S(n, 731)=U(n, 731/2)= S(2*n+1, sqrt(733))/sqrt(733) 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)=731*a(n-1)-a(n-2), n >= 1; a(0)=1, a(1)=731; a(-1):=0.

a(n)=(ap^(n+1) - am^(n+1))/(ap-am) with ap := (731+27*sqrt(733))/2 and am := (731-27*sqrt(733))/2 = 1/ap.

G.f.: 1/(1-731*x+x^2).