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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
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
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
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.)