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 A078986 Chebyshev T(n,19) polynomial. 15
 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 Tanya Khovanova, Recursive Sequences 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(2n*arcsinh(3)). - 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 xrange(0, 16)] # Zerinvary Lajos, Nov 07 2009 (MAGMA) [n: n in [1..10000000] |IsSquare(10*(n^2-1))] // Vincenzo Librandi, Aug 08 2010 (PARI) a(n) = polchebyshev(n, 1, 19); \\ Michel Marcus, Jan 14 2018 CROSSREFS Row 3 of array A188645. Sequence in context: A233011 A280112 A231160 * A180990 A041687 A041684 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 December 18 20:06 EST 2018. Contains 318245 sequences. (Running on oeis4.)