A099140 a(n) = 4^n * T(n,3/2) where T is the Chebyshev polynomial of the first kind.

1, 6, 56, 576, 6016, 62976, 659456, 6905856, 72318976, 757334016, 7930904576, 83053510656, 869747654656, 9108115685376, 95381425750016, 998847258034176, 10460064284409856, 109539215284371456, 1147109554861899776

Offset

0,2

Comments

In general, r^n * T(n,(r+2)/r) has g.f. (1-(r+2)*x)/(1-2*(r+2)*x + r^2*x^2), e.g.f. exp((r+2)*x)*cosh(2*sqrt(r+1)*x), a(n) = Sum_{k=0..n} (r+1)^k*binomial(2n,2k) and a(n) = (1+sqrt(r+1))^(2n)/2 + (1-sqrt(r+1))^(2n)/2.

Links

Harvey P. Dale, Table of n, a(n) for n = 0..900

P. J. Szablowski, On moments of Cantor and related distributions, arXiv preprint arXiv:1403.0386 [math.PR], 2014.

Index entries for linear recurrences with constant coefficients, signature (12,-16).

Formula

G.f.: (1-6*x)/(1-12*x+16*x^2);

E.g.f.: exp(6*x)*cosh(2*sqrt(5)*x);

a(n) = 4^n * T(n, 6/4) where T is the Chebyshev polynomial of the first kind;

a(n) = Sum_{k=0..n} 5^k*binomial(2n, 2k);

a(n) = (1+sqrt(5))^(2n)/2 + (1-sqrt(5))^(2n)/2.

a(n) = a(0)=1, a(1)=6, 12*a(n-1) - 16*a(n-2) for n > 1. - Philippe Deléham, Sep 08 2009

Mathematica

LinearRecurrence[{12, -16}, {1, 6}, 30] (* Harvey P. Dale, Oct 23 2012 *)

Prog

(PARI) a(n) = 4^n*polchebyshev(n, 1, 3/2); \\ Michel Marcus, Sep 08 2019

Crossrefs

Cf. A001541, A081294, A083884, A090965, A099141, A099142.

Sequence in context: A181589 A093142 A092655 * A371739 A048348 A227384

Adjacent sequences: A099137 A099138 A099139 * A099141 A099142 A099143

Keyword

easy,nonn

Author

Paul Barry, Sep 30 2004

Status

approved