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A078987 Chebyshev U(n,x) polynomial evaluated at x=19. 9
1, 38, 1443, 54796, 2080805, 79015794, 3000519367, 113940720152, 4326746846409, 164302439443390, 6239165952002411, 236924003736648228, 8996872976040630253, 341644249085807301386, 12973484592284636822415, 492650770257730391950384, 18707755785201470257292177 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

A078986(n+1)^2 - 10*(6*a(n))^2 = +1, n>=0 (Pell equation +1, see A033313 and A033317).

a(n) equals the number of 01-avoiding words of length n on alphabet {0,1,...,37}. - Milan Janjic, Jan 26 2015

LINKS

Colin Barker, Table of n, a(n) for n = 0..632

R. Flórez, R. A. Higuita, A. Mukherjee, Alternating Sums in the Hosoya Polynomial Triangle, Article 14.9.5 Journal of Integer Sequences, Vol. 17 (2014).

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), n>=1, a(-1)=0, a(0)=1.

a(n) = S(n, 38) with S(n, x) = U(n, x/2), Chebyshev's polynomials of the second kind. See A049310.

G.f.: 1/(1-38*x+x^2).

a(n) = Sum_{k=0..floor(n/2)} (-1)^k*binomial(n-k, k)*38^(n-2*k).

a(n) = ((19+6*sqrt(10))^(n+1) - (19-6*sqrt(10))^(n+1))/(12*sqrt(10)).

a(n) = Sum_{k=0..n} A101950(n,k)*37^k. - Philippe Deléham, Feb 10 2012

Product_{n>=0} (1 + 1/a(n)) = 1/3*(3 + sqrt(10)). - Peter Bala, Dec 23 2012

Product_{n>=1} (1 - 1/a(n)) = 3/19*(3 + sqrt(10)). - Peter Bala, Dec 23 2012

MATHEMATICA

lst={}; Do[AppendTo[lst, GegenbauerC[n, 1, 19]], {n, 0, 8^2}]; lst (* Vladimir Joseph Stephan Orlovsky, Sep 11 2008 *)

PROG

(Sage) [lucas_number1(n, 38, 1) for n in xrange(1, 16)] # Zerinvary Lajos, Nov 07 2009

(PARI) a(n)=subst(polchebyshev(n, 2), x, 19) \\ Charles R Greathouse IV, Feb 10 2012

(PARI) Vec(1/(1-38*x+x^2) + O(x^50)) \\ Colin Barker, Jun 15 2015

CROSSREFS

Sequence in context: A218740 A158702 A239364 * A009982 A041685 A221385

Adjacent sequences:  A078984 A078985 A078986 * A078988 A078989 A078990

KEYWORD

nonn,easy

AUTHOR

Wolfdieter Lang, Jan 10 2003

STATUS

approved

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Last modified March 25 05:37 EDT 2017. Contains 284036 sequences.