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 A078986 Chebyshev T(n,19) polynomial. 18
 1, 19, 721, 27379, 1039681, 39480499, 1499219281, 56930852179, 2161873163521, 82094249361619, 3117419602578001, 118379850648602419, 4495316905044313921, 170703662541035326579, 6482243859654298096081, 246154563004322292324499, 9347391150304592810234881, 354954709148570204496600979, 13478931556495363178060602321 (list; graph; refs; listen; history; text; internal format)
 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

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Last modified July 23 15:54 EDT 2024. Contains 374552 sequences. (Running on oeis4.)