A098248 Chebyshev polynomials S(n,291).

1, 291, 84680, 24641589, 7170617719, 2086625114640, 607200737742521, 176693328057958971, 51417151264128318040, 14962214324533282590669, 4353952951287921105566639, 1266985346610460508437301280

Offset: 0

Author

Wolfdieter Lang, Sep 10 2004

Keywords

• nonn
• easy

Comments

Used for all positive integer solutions of Pell equation x^2 - 293*y^2 = -4. See A098249 with A098250.

Links

• Indranil Ghosh, Table of n, a(n) for n = 0..405
• Tanya Khovanova, Recursive Sequences
• Index entries for sequences related to Chebyshev polynomials.
• Index entries for linear recurrences with constant coefficients, signature (291,-1).

Programs

Mathematica

LinearRecurrence[{291,-1},{1,291},20] (* Harvey P. Dale, Dec 27 2015 *)

Formula

a(n)= S(n, 291)=U(n, 291/2)= S(2*n+1, sqrt(293))/sqrt(293) 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)=291*a(n-1)-a(n-2), n >= 1; a(0)=1, a(1)=291; a(-1):=0.

a(n)=(ap^(n+1) - am^(n+1))/(ap-am) with ap := (291+17*sqrt(293))/2 and am := (291-17*sqrt(293))/2 = 1/ap.

G.f.: 1/(1-291*x+x^2).