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%I #55 Feb 07 2023 02:40:44
%S 1,2,8,19,42,78,139,224,350,517,744,1032,1405,1862,2432,3115,3942,
%T 4914,6067,7400,8954,10729,12768,15072,17689,20618,23912,27571,31650,
%U 36150,41131,46592,52598,59149,66312,74088,82549,91694,101600,112267,123774
%N Number of different cuboids with volume (pq)^n, where p,q are distinct prime numbers.
%C Subsequence of A034836, which gives the number of cuboids for volume n.
%H Alois P. Heinz, <a href="/A101427/b101427.txt">Table of n, a(n) for n = 0..10000</a>
%H Geoffrey B. Campbell, <a href="https://arxiv.org/abs/2302.01091">Vector Partition Identities for 2D, 3D and nD Lattices</a>, arXiv:2302.01091 [math.CO], 2023.
%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,-3,-1,1,3,-1,-2,1).
%F If n is a multiple of 3, a(n) = ((n+2)^2*(n+1)^2 + 12*(floor(n/2)+1)^2+8)/24, otherwise a(n) = ((n+2)^2*(n+1)^2 + 12*(floor(n/2)+1)^2)/24. - _Frederic Solbes_, Mar 18 2014
%F G.f.: -(x^6+3*x^4+4*x^3+3*x^2+1)/((x^2+x+1)*(x+1)^2*(x-1)^5). - _Colin Barker_, Mar 27 2014
%F From _Daniel Mondot_, Sep 20 2016: (Start)
%F a(n) = a(n-1) + 2*a(n-2) - a(n-3) - 2*a(n-4) - a(n-5) + 2*a(n-6) + a(n-7) - a(n-8) + 12, n>=8.
%F a(n) = 4*a(n-6) - 6*a(n-12) + 4*a(n-18) - a(n-24) + 1296, n>=24. (End)
%t a[n_] := Switch[Mod[n, 6], 0, n+1, 1|5, 3n/4 + 7/24, 2|4, n+2/3, 3, 3n/4 + 5/8] + n^4/24 + n^3/4 + 2n^2/3; Table[a[n], {n, 0, 40}] (* _Jean-François Alcover_, Oct 06 2016, after _Frederic Solbes_' formula *)
%o (PARI) a(n) = if (n % 3, ((n+2)^2*(n+1)^2 + 12*(n\2+1)^2)/24, ((n+2)^2*(n+1)^2 + 12*(n\2+1)^2+8)/24); \\ _Michel Marcus_, Mar 18 2014
%Y Cf. A034836, A101423, A101424, A101425, A101426.
%Y Column k=3 of A277239.
%K nonn
%O 0,2
%A _Anthony C Robin_, Jan 17 2005
%E Extended by _Ray Chandler_, Dec 17 2008
%E Edited by _Ray Chandler_, Dec 19 2008
%E a(0) = 1 prepended by _Daniel Mondot_, Sep 20 2016
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