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A384511 a(n) is the number of ways to partition n X n X n cube into five distinct cuboids with three full-length axial spanning parts sharing only two cube corners each.

%I A384511 #19 Jun 18 2025 19:20:25
%S A384511 0,0,1,3,10,18,35,53,84,116,165,215,286,358,455,553,680,808,969,1131,
%T A384511 1330,1530,1771,2013,2300,2588,2925,3263,3654,4046,4495,4945,5456,
%U A384511 5968,6545,7123,7770,8418,9139,9861,10660,11460,12341,13223,14190
%N A384511 a(n) is the number of ways to partition n X n X n cube into five distinct cuboids with three full-length axial spanning parts sharing only two cube corners each.
%C A384511 Alternatively a(n) is the number of distinct sets of five unordered triplets of distinct element composition generated by (x,n,z), (n,y,n-z), (n-x,n-y,n), (n-x,y,z), (x,n-y,n-z), where 0 < x,y,z < n.
%C A384511 Three elements in a triplet representing the three dimensions of a cuboid and exactly three of the five cuboids span through the entire length n along one axis, connecting opposite faces of the cube while sharing only two of their corners with the cube.
%H A384511 Janaka Rodrigo, <a href="/A384511/a384511.pdf">Distinct Five-Cuboid Combinations in Triplets Form</a>
%e A384511 Triplet (3,3,3) can be decomposed by the rule in only one way:
%e A384511   {(1,3,1), (3,1,2), (2,2,3), (2,1,1), (1,2,2)}.
%e A384511 Therefore, a(3) = 1.
%e A384511 Triplet (4,4,4) can be decomposed by the rule in only three different ways:
%e A384511   {(1,4,1), (4,1,3), (3,3,4), (3,1,1), (1,3,3)},
%e A384511   {(1,4,2), (4,2,2), (3,2,4), (3,2,2), (1,2,2)},
%e A384511   {(1,4,3), (4,2,1), (3,2,4), (3,2,3), (1,2,1)}.
%e A384511 Therefore, a(4) = 3.
%Y A384511 Cf. A384479.
%K A384511 nonn
%O A384511 1,4
%A A384511 _Janaka Rodrigo_, May 31 2025