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

0,2

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).

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

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 *)

Crossrefs

Cf. A002965.

Sequence in context: A348602 A270849 A324454 * A006243 A171391 A151738

Adjacent sequences: A097728 A097729 A097730 * A097732 A097733 A097734

Keyword

nonn,easy

Author

Wolfdieter Lang, Aug 31 2004

Status

approved