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A168503
Expansion of 1/(1-x/(1-2x/(1-3x/(1-x/(1-2x/(1-3x/(1-... (continued fraction).
1
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
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 A359797
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Nov 27 2009
STATUS
approved