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%I #20 Feb 13 2024 19:23:56
%S 0,1,4,24,56,128,384,832,1792,4608,9728,20480,49152,102400,212992,
%T 491520,1015808,2097152,4718592,9699328,19922944,44040192,90177536,
%U 184549376,402653184,822083584,1677721600,3623878656,7381975040,15032385536,32212254720
%N a(n) = 2^(n-1)*A047240(n).
%C -a(n) is the Hankel transform of A030662(n) = binomial(2*n,n)-1.
%H Colin Barker, <a href="/A128205/b128205.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,8,-16).
%F a(n) = 2^(n-1)*(cos(2*Pi*n/3) + sqrt(3)*sin(2*Pi*n/3)/3 + 2n - 1);
%F O.g.f.: x(1+2x+16x^2)/((2x-1)^2*(4x^2+2x+1)). a(n) = 2a(n-1) + 8a(n-3) - 16a(n-4). - _R. J. Mathar_, Apr 28 2008
%t a047240[n_] := 6 Floor[n/3] + Mod[n, 3]
%t a128205[n_] := Map[2^(#-1) a047240[#]&, Range[0, n]]
%t a128205[25] (* data *) (* _Hartmut F. W. Hoft_, Mar 13 2017 *)
%t LinearRecurrence[{2,0,8,-16},{0,1,4,24},40] (* _Harvey P. Dale_, Feb 13 2024 *)
%o (PARI) concat(0, Vec(x*(1 + 2*x + 16*x^2) / ((1 - 2*x)^2*(1 + 2*x + 4*x^2)) + O(x^40))) \\ _Colin Barker_, Mar 13 2017
%Y Cf. A047240, A030662.
%K easy,nonn
%O 0,3
%A _Paul Barry_, Feb 19 2007
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