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%I #18 Sep 11 2022 10:09:17
%S 1,4,9,32,125,36,343,256,243,500,1331,288,2197,1372,1125,2048,4913,
%T 972,6859,4000,3087,5324,12167,2304,15625,8788,6561,10976,24389,4500,
%U 29791,16384,11979,19652,42875,7776,50653,27436,19773,32000,68921,12348,79507,42592
%N a(n) = denominator(3*(3+(-1)^n)/(n+1)^3).
%C Numerator of 3*(3+(-1)^n)/(n+1)^3 is A129197.
%C (1/(2*Pi))*int(exp(i*(n+1)*t)((t-Pi)/i)^3,t,0,2*Pi)) = (A129202(n)*Pi^2-A129203(n))/A129196(n) with i=sqrt(-1).
%H <a href="/index/Rec#order_24">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,0,0,4,0,0,0,0,0,-6,0,0,0,0,0,4,0,0,0,0,0,-1).
%F a(n) = A129204(n+1)/(5/3+(4/3)*cos(2*Pi*(n+1)/3)).
%F a(n) = denominator((1/(2*Pi))*Integral_{t=0..2*Pi} (exp(i*(n+1)*t)((t-Pi)/i)^3)) with i=sqrt(-1).
%F a(n) = denominator((Pi^2*(n+1)^2-6)/(n+1)^3).
%F a(n) = ((n+1)^3/(gcd(n+1,2)*gcd(n+1,3))). - _Paul Barry_, Oct 09 2007
%F a(n) = numerator of coefficient of x^6 in the Maclaurin expansion of -exp(-(n+1)*x^2). - _Francesco Daddi_, Aug 04 2011
%F Sum_{n>=0} 1/a(n) = 29*zeta(3)/24. - _Amiram Eldar_, Sep 11 2022
%t a[n_] := Denominator[3*(3 + (-1)^n)/(n + 1)^3]; Array[a, 50, 0] (* _Amiram Eldar_, Sep 11 2022 *)
%Y Cf. A129196, A129197, A129202, A129203, A129204.
%K nonn,frac,easy
%O 0,2
%A _Paul Barry_, Apr 02 2007, Apr 03 2007
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