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Sequence with Hankel transform equal to 3^floor(n^2/3).
2

%I #13 Sep 08 2013 13:31:50

%S 1,1,2,7,32,160,830,4405,23798,130498,724748,4069258,23064608,

%T 131809108,758696492,4394825647,25600773272,149877922228,881394158558,

%U 5204245242208,30841413359186,183381577399006,1093695670905206

%N Sequence with Hankel transform equal to 3^floor(n^2/3).

%C Hankel transform is A168495.

%H Vincenzo Librandi, <a href="/A168494/b168494.txt">Table of n, a(n) for n = 0..300</a>

%F G.f.: 1/(1-x/(1-x/(1-3x/(1-x/(1-x/(1-3x/(1-x/(1-x/(1-3x/(1-.... (continued fraction);

%F G.f.: 1/(1-x-x^2/(1-4x-3x^2/(1-2x-3x^2/(1-4x-x^2/(1-4x-3x^2/(1-2x-3x^2/(1-4x-x^2/(1-... (continued fraction),

%F with sequences (1,3,3,1,3,3,1,3,3,1,...) and (1,4,2,4,4,2,4,4,2,4,4,...).

%F G.f.: (1+x-sqrt(1-10x+25x^2-12x^3))/(6x(1-x)).

%F a(n) = Sum_{k=0..n} A091866(n,k)*3^(n-k). - _Philippe Deléham_, Nov 27 2009

%F Conjecture: (n+1)*a(n) +(4-11*n)*a(n-2) +5*(7*n-11)*a(n-2) +(92-37*n) * a(n-3) +6*(2*n-7)*a(n-4) = 0. - _R. J. Mathar_, Sep 30 2012

%F a(n) ~ sqrt(33-sqrt(33))*((7+sqrt(33))/2)^n/(12*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Oct 24 2012

%t CoefficientList[Series[(1+x-Sqrt[1-10*x+25*x^2-12*x^3])/(6*x*(1-x)), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Oct 24 2012 *)

%Y Cf. A000108, A091866, A109033. - _Philippe Deléham_, Nov 27 2009

%K easy,nonn

%O 0,3

%A _Paul Barry_, Nov 27 2009