A098257 Chebyshev polynomials S(n,531).

1, 531, 281960, 149720229, 79501159639, 42214966048080, 22416067470370841, 11902889611800868491, 6320411967798790797880, 3356126852011546112805789, 1782097038006163187109076079, 946290171054420640808806592160

Offset: 0

Author

Wolfdieter Lang, Sep 10 2004

Keywords

• nonn
• easy

Comments

Used for all positive integer solutions of Pell equation x^2 - 533*y^2 = -4. See A098258 with A098259.

Links

• Tanya Khovanova, Recursive Sequences
• Index entries for linear recurrences with constant coefficients, signature (531, -1).
• Index entries for sequences related to Chebyshev polynomials.

Formula

a(n)= S(n, 531)=U(n, 531/2)= S(2*n+1, sqrt(533))/sqrt(533) 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)=531*a(n-1)-a(n-2), n >= 1; a(0)=1, a(1)=531; a(-1):=0.

a(n)=(ap^(n+1) - am^(n+1))/(ap-am) with ap := (531+23*sqrt(533))/2 and am := (531-23*sqrt(533))/2 = 1/ap.

G.f.: 1/(1-531*x+x^2).