A097731 Chebyshev U(n,x) polynomial evaluated at x=99 gives 2*7^2+1.

1, 198, 39203, 7761996, 1536836005, 304285766994, 60247045028807, 11928610629936792, 2361804657682456009, 467625393610496352990, 92587466130220595436011, 18331850668390067399977188, 3629613844875103124600047213, 718645209434602028603409370986

Offset: 0

Author

Wolfdieter Lang, Aug 31 2004

Keywords

• nonn
• easy

Comments

Used to form integer solutions of Pell equation a^2 - 50*b^2 =-1. See A097732 with A097733.

Links

• Indranil Ghosh, Table of n, a(n) for n = 0..434
• 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 (198,-1).

Crossrefs

Cf. A002965.

Programs

Maple

with(combinat): seq(fibonacci(6*n+6,2)/70, n=0..12); # Zerinvary Lajos, Apr 21 2008

Mathematica

LinearRecurrence[{198, -1},{1, 198},12] (* Ray Chandler, Aug 11 2015 *)

Formula

a(n) = 2*99*a(n-1) - a(n-2), n>=1, a(0)=1, a(-1):=0.

a(n) = S(n, 2*99)= U(n, 99), Chebyshev's polynomials of the second kind. See A049310.

G.f.: 1/(1-198*x+x^2).

a(n) = sum((-1)^k*binomial(n-k, k)*198^(n-2*k), k=0..floor(n/2)), n>=0.

a(n) = ((99+70*sqrt(2))^(n+1) - (99-70*sqrt(2))^(n+1))/(140*sqrt(2)), n>=0.

a(n) = Pell(6*n + 6)/Pell(6). Sum_{n >= 0} 1/( 14*a(n) + 1/(14*a(n)) ) = 1/14. - Peter Bala, Mar 25 2015

a(n) = A002965(12*(n+1))/70. - Gerry Martens, Jul 14 2023