A078987 Chebyshev U(n,x) polynomial evaluated at x=19.

1, 38, 1443, 54796, 2080805, 79015794, 3000519367, 113940720152, 4326746846409, 164302439443390, 6239165952002411, 236924003736648228, 8996872976040630253, 341644249085807301386, 12973484592284636822415, 492650770257730391950384, 18707755785201470257292177

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

Hacène Belbachir, Soumeya Merwa Tebtoub, and László Németh, Ellipse Chains and Associated Sequences, J. Int. Seq., Vol. 23 (2020), Article 20.8.5.

R. Flórez, R. A. Higuita, and 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

From Andrea Pinos, Jan 02 2023: (Start)

a(n) = (A097314(n+1) - A097315(n+1))/2.

a(n) = (A097314(n) + A097315(n))/2. (End)

Maple

seq( simplify(ChebyshevU(n, 19)), n=0..20); # G. C. Greubel, Dec 22 2019

Mathematica

lst={}; Do[AppendTo[lst, GegenbauerC[n, 1, 19]], {n, 0, 8^2}]; lst (* Vladimir Joseph Stephan Orlovsky, Sep 11 2008 *)
ChebyshevU[Range[21] -1, 19] (* G. C. Greubel, Dec 22 2019 *)

Prog

(Sage) [lucas_number1(n, 38, 1) for n in range(1, 16)] # Zerinvary Lajos, Nov 07 2009
(Sage) [chebyshev_U(n, 19) for n in (0..20)] # G. C. Greubel, Dec 22 2019
(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
(Magma) m:=19; I:=[1, 2*m]; [n le 2 select I[n] else 2*m*Self(n-1) -Self(n-2): n in [1..20]]; // G. C. Greubel, Dec 22 2019
(GAP) m:=19;; a:=[1, 2*m];; for n in [3..20] do a[n]:=2*m*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Dec 22 2019

Crossrefs

Cf. A097314, A097315.

Chebyshev sequence U(n, m): A000027 (m=1), A001353 (m=2), A001109 (m=3), A001090 (m=4), A004189 (m=5), A004191 (m=6), A007655 (m=7), A077412 (m=8), A049660 (m=9), A075843 (m=10), A077421 (m=11), A077423 (m=12), A097309 (m=13), A097311 (m=14), A097313 (m=15), A029548 (m=16), A029547 (m=17), A144128 (m=18), this sequence (m=19), A097316 (m=33).

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