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A078986
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Chebyshev T(n,19) polynomial.
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6
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1, 19, 721, 27379, 1039681, 39480499, 1499219281, 56930852179, 2161873163521, 82094249361619, 3117419602578001, 118379850648602419, 4495316905044313921, 170703662541035326579, 6482243859654298096081
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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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 (benoit7848c(AT)orange.fr), Feb 14 2004
Numbers n such that 10*(n^2-1) is a square. [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Aug 08 2010]
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LINKS
| Tanya Khovanova, Recursive Sequences
Index entries for sequences related to Chebyshev polynomials.
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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(((-1)^k)*(n/(2*(n-k)))*binomial(n-k, k)*(2*19)^(n-2*k), k=0..floor(n/2)), n>=1.
a(n) = Cosh[2n*ArcSinh[3]] - Herbert Kociemba (kociemba(AT)t-online.de), Apr 24 2008
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PROG
| (Other) sage: [lucas_number2(n, 38, 1)/2 for n in xrange(0, 16)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Nov 07 2009]
(MAGMA) [n: n in [1..10000000] |IsSquare(10*(n^2-1))] [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Aug 08 2010]
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CROSSREFS
| Row 3 of array A188645.
Sequence in context: A107118 A200849 A162012 * A180990 A041687 A041684
Adjacent sequences: A078983 A078984 A078985 * A078987 A078988 A078989
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KEYWORD
| nonn,easy
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AUTHOR
| Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Jan 10 2003
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