A078986 Chebyshev T(n,19) polynomial.

1, 19, 721, 27379, 1039681, 39480499, 1499219281, 56930852179, 2161873163521, 82094249361619, 3117419602578001, 118379850648602419, 4495316905044313921, 170703662541035326579, 6482243859654298096081, 246154563004322292324499, 9347391150304592810234881, 354954709148570204496600979, 13478931556495363178060602321

Offset: 0

Author

Wolfdieter Lang, Jan 10 2003

Keywords

• nonn
• easy

Comments

a(n+1)^2 - 10*(6*A078987(n))^2 = 1, n >= 0 (Pell equation +1, see A033313 and A033317).

Also gives solutions to the equation x^2 - 1 = floor(x*r*floor(x/r)) where r=sqrt(10). - Benoit Cloitre, Feb 14 2004

Numbers n such that 10*(n^2 - 1) is a square. - Vincenzo Librandi, Aug 08 2010

Links

• Indranil Ghosh, Table of n, a(n) for n = 0..632
• Hacène Belbachir, Soumeya Merwa Tebtoub, and László Németh, Ellipse Chains and Associated Sequences, J. Int. Seq., Vol. 23 (2020), Article 20.8.5.
• Tanya Khovanova, Recursive Sequences
• Index entries for sequences related to Chebyshev polynomials.
• Index entries for linear recurrences with constant coefficients, signature (38,-1).

Crossrefs

Row 3 of array A188645.

Programs

Mathematica

LinearRecurrence[{38, -1},{1, 19},15] (* Ray Chandler, Aug 11 2015 *)

PARI

a(n) = polchebyshev(n, 1, 19); \\ Michel Marcus, Jan 14 2018

Sage

[lucas_number2(n,38,1)/2 for n in range(0, 16)] # Zerinvary Lajos, Nov 07 2009

Formula

a(n) = 38*a(n-1) - a(n-2), a(-1) := 19, a(0)=1.

G.f.: (1-19*x)/(1-38*x+x^2).

a(n) = T(n, 19) = (S(n, 38)-S(n-2, 38))/2 = S(n, 38)-19*S(n-1, 38) with T(n, x), resp. S(n, x), Chebyshev's polynomials of the first, resp. second, kind. See A053120 and A049310. S(n, 38) = A078987(n).

a(n) = (ap^n + am^n)/2 with ap := 19+6*sqrt(10) and am := 19-6*sqrt(10).

a(n) = Sum_{k=0..floor(n/2)} ((-1)^k)*(n/(2*(n-k)))*binomial(n-k, k)*(2*19)^(n-2*k), n >= 1.

a(n) = cosh(2*arcsinh(3)*n). - Herbert Kociemba, Apr 24 2008

Extensions

More terms from Indranil Ghosh, Feb 04 2017