OFFSET
0,3
COMMENTS
Hankel transform is A168504. First column of array whose production matrix begins
1, 1,
2, 5, 1,
0, 3, 3, 1,
0, 0, 6, 4, 1,
0, 0, 0, 2, 5, 1,
0, 0, 0, 0, 3, 3, 1
0, 0, 0, 0, 0, 6, 4, 1
FORMULA
G.f.: 1/(1-x-2x^2/(1-5x-3x^2/(1-3x-6x^2/(1-4x-2x^2/(1-5x-3x^2/(1-3x-6x^2/(1-... (continued fraction, defined by the sequences (1,5,3,4,5,3,4,5,3,4,....) and (2,3,6,2,3,6,2,3,6,...);
G.f.: (1-sqrt(1-12*x+36*x^2-24*x^3))/(6*x*(1-x)).
G.f.: (2*x-1)/(Q(0)-3*x), where Q(k) = 6*x - 1 - 6*x^3/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Sep 26 2013
Conjecture: (n+1)*a(n) +(-13*n+5)*a(n-1) +6*(8*n-13)*a(n-2) +12*(-5*n+13)*a(n-3) +12*(2*n-7)*a(n-4)=0. - R. J. Mathar, Feb 10 2015
a(n) ~ sqrt(3*s - sqrt(3)) * (3 - 4*s^2) * 2^(2*n + 3) * s^(n + 5/2) * (sqrt(3) + 2*s)^n / (3 * sqrt(Pi) * n^(3/2)), where s = sin(2*Pi/9). - Vaclav Kotesovec, Jun 06 2022
MATHEMATICA
Join[{1}, Table[SeriesCoefficient[Series[1/(1+ContinuedFractionK[(Mod[k-1, 3]+1)*x*-1, 1, {k, 1, 50}]), {x, 0, 50}], n], {n, 1, 50}]] (* Benedict W. J. Irwin, Feb 07 2016 *)
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Nov 27 2009
STATUS
approved