A097728 Chebyshev U(n,x) polynomial evaluated at x=73 = 2*6^2+1.

1, 146, 21315, 3111844, 454307909, 66325842870, 9683118751111, 1413669011819336, 206385992606871945, 30130941251591484634, 4398911036739749884619, 642210880422751891669740

Offset: 0

Author

Wolfdieter Lang, Aug 31 2004

Keywords

• nonn
• easy

Comments

Used to form integer solutions of Pell equation a^2 - 37*b^2 =-1. See A097729 with A097730.

Links

• Indranil Ghosh, Table of n, a(n) for n = 0..461
• R. Flórez, R. A. Higuita, A. Mukherjee, Alternating Sums in the Hosoya Polynomial Triangle, Article 14.9.5 Journal of Integer Sequences, Vol. 17 (2014).
• Tanya Khovanova, Recursive Sequences
• Index entries for sequences related to Chebyshev polynomials.
• Index entries for linear recurrences with constant coefficients, signature (146, -1)

Programs

Mathematica

LinearRecurrence[{146, -1},{1, 146},12] (* Ray Chandler, Aug 11 2015 *)

Formula

a(n) = 2*73*a(n-1) - a(n-2), n>=1, a(0)=1, a(-1):=0.

a(n) = S(n, 2*73)= U(n, 73), Chebyshev's polynomials of the second kind. See A049310.

G.f.: 1/(1-146*x+x^2).

a(n)= sum((-1)^k*binomial(n-k, k)*146^(n-2*k), k=0..floor(n/2)), n>=0.

a(n) = ((73+12*sqrt(37))^(n+1) - (73-12*sqrt(37))^(n+1))/(24*sqrt(37)).