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Alternatively, a(n) is the number of ways to decompose the triplet (n,n,n) into 4 distinct unordered triplets.

Start with initial triplet (n,n,n). At each step choose a triplet from the current set and apply the rule (x,y,z) -> (x,y,z-r) and (x,y,r) for 0 < r <= z/2 (or similarly on x or y), checking to ensure the new triplets are distinct within the set.

If two duplicates (triplets with same element composition) appear, mark one for further decomposition in the next step.

Continue until reaching a set of exactly four triplets, all with distinct element composition, and with total volume (sum of the products of elements in each triplet) = n^3.

Partitioning n X n X n cube is done by decomposing the triplet (n,n,n) into five unordered integer triplets of distinct element composition in three different stages and applying the inclusion-exclusion principle to obtain all geometrically feasible triplets without repetitions.

First stage:

Generating the sequence of sets A(n) of five triplets of distinct element composition by decomposing the triplet (n,n,n).

The number of sets in each term describes the number of ways a cube can be partitioned into five cuboids by cutting one of the cuboids in the previous stage into two cuboids. The algorithm explained in A381847 (partitioning into three cuboids) and A384311 (partitioning into four cuboids). As an example: A(3) = {{(3,3,1), (3,2,2), (3,1,1), (2,1,1), (1,1,1)}, {(3,2,2), (3,2,1), (2,2,1), (3,1,1), (2,1,1)}}.

Therefore the number of sets in A(3) is |A(3)| = 2.

Second stage:

Generating the sequence of sets B(n) of five triplets of distinct element composition by decomposing the triplet (n,n,n).

The number of sets in each term represents number of distinct five-cuboid combinations filling the n X n X n cube with three full-length axial spanning sharing only two cube corners each.

Sets of five distinct triplets of the n-th term are defined by {(n,n-x,y), (n,n-y,z), (n,n-z,x), (x,y,z), (n-x,n-y,n-z)}, where 0 < x,y,z < n.

The triplet (3,3,3) can be decomposed by the rule only in one way giving, B(3) = {{(1,3,1), (3,1,2), (2,2,3), (2,1,1), (1,2,2)}}. Therefore |B(3)| = 1.

Third stage:

Generating the sequence of sets C(n) of five triplets of distinct element composition by decomposing the triplet (n,n,n).

The number of sets in each term represents number of different ways to partition n X n X n cube into five distinct cuboids such that all five cuboids going across the cube parallel to each other avoiding any cut-plane that cuts through the whole pile.

Sets of five distinct triplets of the n-th term are defined by {(n,x,y), (n,p,y+q), (n,n-p,q), (n,n-p-x,n-q), (n,p+x,n-y-q)}, where p+x < n,q+y < n and 0 < x,y,p,q < n.

Triplets (1,1,1), (2,2,2) and (3,3,3) cannot be decomposed by this rule and the triplet (4,4,4) has only one way of decomposing, C(4) = {{(1,1,4), (1,2,4), (1,3,4), (2,2,4), (2,3,4)}}. Therefore |C(4)| = 1.

Since there are no intersections between B(n) and C(n), the number of ways to partition n X n X n cube is given by: a(n) = |A(n) union B(n) union C(n)| = |A(n)| + |B(n)| + |C(n)| - |A(n) intersection B(n)| - |A(n) intersection C(n)|.

Alternatively, a(n) is the number of ways to decompose the triplet (n,n,n) into 3 distinct unordered triplets.

Initial stage: One triplet of the form (n,n,n).

Second stage: Decompose the original triplet into two distinct triplets by splitting one of the elements of (x,y,z) into two parts at a time according to the following rule; (x,y,z) is replaced by (x,y,z-r), (x,y,r), where 0 < r <= z/2. Each resulting triplet must be distinct in element composition when comparing with the rest of the triplets in the set. Sets including the same element composition including the case r = z/2 are named as duplicates and set aside to reconsider when forming the next term.

Third stage: Apply the same rule to one of the two triplets of the second term at a time to create another two distinct triplets. At this stage consider the duplicates of the second term and apply the same rule to decompose one of the two identical triplets into two triplets of distinct element composition.