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

0,2

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).

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

Mathematica

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

Prog

(Sage) [lucas_number2(n, 38, 1)/2 for n in range(0, 16)] # Zerinvary Lajos, Nov 07 2009
(PARI) a(n) = polchebyshev(n, 1, 19); \\ Michel Marcus, Jan 14 2018

Crossrefs

Row 3 of array A188645.

Sequence in context: A280112 A231160 A344131 * A364299 A180990 A041687

Adjacent sequences: A078983 A078984 A078985 * A078987 A078988 A078989

Keyword

nonn,easy

Author

Wolfdieter Lang, Jan 10 2003

Extensions

More terms from Indranil Ghosh, Feb 04 2017

Status

approved