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A386884 a(n) is the number of distinct four-cuboid combinations that fill an n X n X n cube using only strict cuboids.

%I A386884 #9 Aug 08 2025 16:27:05
%S A386884 0,0,0,0,1,2,9,12,30,36,70,80,135,150,231,252,364,392,540,576,765,810,
%T A386884 1045,1100,1386,1452,1794,1872,2275,2366,2835,2940,3480,3600,4216,
%U A386884 4352,5049,5202,5985,6156,7030,7220,8190,8400,9471,9702,10879,11132,12420
%N A386884 a(n) is the number of distinct four-cuboid combinations that fill an n X n X n cube using only strict cuboids.
%C A386884 A strict cuboid is a cuboid with all three dimensions different.
%C A386884 Alternatively a(n) is the number of ways to decompose triplet (n,n,n) into sets of distinct four unordered triplets of the form (x,y,z) with x != y != z for each of the four triplets.
%e A386884 As described in A384311 there are 85 sets of distinct four-cuboid combinations filling 6 X 6 X 6 cube and only two of those have all four triplets with different elements, those are;
%e A386884    {(1,2,6), (1,4,6), (2,5,6), (4,5,6)},
%e A386884    {(1,3,6), (2,3,6), (3,4,6), (3,5,6)}.
%e A386884 Therefore a(6) = 2.
%Y A386884 Cf. A384311.
%K A386884 nonn
%O A386884 1,6
%A A386884 _Janaka Rodrigo_, Aug 06 2025
%E A386884 More terms from _Sean A. Irvine_, Aug 06 2025