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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. [From Vincenzo Librandi, Aug 08 2010]

LINKS

Indranil Ghosh, Table of n, a(n) for n = 0..632

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(((-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, 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)]# [From Zerinvary Lajos, Nov 07 2009]

(MAGMA) [n: n in [1..10000000] |IsSquare(10*(n^2-1))] [From Vincenzo Librandi, Aug 08 2010]

CROSSREFS

Row 3 of array A188645.

Sequence in context: A223521 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 June 27 19:44 EDT 2017. Contains 288790 sequences.