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.

0, 0, 1, 3, 10, 18, 35, 53, 84, 116, 165, 215, 286, 358, 455, 553, 680, 808, 969, 1131, 1330, 1530, 1771, 2013, 2300, 2588, 2925, 3263, 3654, 4046, 4495, 4945, 5456, 5968, 6545, 7123, 7770, 8418, 9139, 9861, 10660, 11460, 12341, 13223, 14190

Offset: 1

Author

Janaka Rodrigo, May 31 2025

Keywords

• nonn

Comments

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.

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.

Examples

			Triplet (3,3,3) can be decomposed by the rule in only one way:
  {(1,3,1), (3,1,2), (2,2,3), (2,1,1), (1,2,2)}.
Therefore, a(3) = 1.
Triplet (4,4,4) can be decomposed by the rule in only three different ways:
  {(1,4,1), (4,1,3), (3,3,4), (3,1,1), (1,3,3)},
  {(1,4,2), (4,2,2), (3,2,4), (3,2,2), (1,2,2)},
  {(1,4,3), (4,2,1), (3,2,4), (3,2,3), (1,2,1)}.
Therefore, a(4) = 3.
		

Links

• Janaka Rodrigo, Distinct Five-Cuboid Combinations in Triplets Form

Crossrefs

Cf. A384479.