A168503 Expansion of 1/(1-x/(1-2x/(1-3x/(1-x/(1-2x/(1-3x/(1-... (continued fraction).

1, 1, 3, 15, 87, 531, 3339, 21483, 140859, 938331, 6334875, 43257483, 298276587, 2074128363, 14529077163, 102432060459, 726280074027, 5175707802795, 37051160719275, 266319772644267, 1921345252699563, 13907901060055467

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

Links

Table of n, a(n) for n=0..21.

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

Sequence in context: A152596 A278392 A370287 * A370184 A089022 A394121

Adjacent sequences: A168500 A168501 A168502 * A168504 A168505 A168506

Keyword

easy,nonn

Author

Paul Barry, Nov 27 2009

Status

approved