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

1,4

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.

Links

Table of n, a(n) for n=1..45.

Janaka Rodrigo, Distinct Five-Cuboid Combinations in Triplets Form

Example

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.

Crossrefs

Cf. A384479.

Sequence in context: A210286 A275988 A177955 * A372608 A298976 A265487

Adjacent sequences: A384508 A384509 A384510 * A384512 A384513 A384514

Keyword

nonn

Author

Janaka Rodrigo, May 31 2025

Status

approved