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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)|.

The partitions here must be valid packings of the n X n X n cube, hence T(n,k) is generally less than the number of partitions of n^3 into distinct cuboids (x,y,z) with 1 <= x,y,z <= n and volume x*y*z.

Alternatively a(n) is the total number of distinct sets of five unordered integer duplets with distinct element composition of the form: (x,y), (p,y+q), (n-p,q), (n-p-x,n-q), (p+x,n-y-q) where elements of a duplet represent the lengths of the two sides of a rectangle, p+x < n, q+y < n and 0 < x,y,p,q < n.

A386296 is the main sequence for this topic.

Alternatively a(n) is the number of distinct six-cuboid combinations filling an n X n X n cube.

Alternatively a(n) is the number of distinct five-triplet sets produced by A(n)-D(n); that is, a(n) = |A(n)-D(n)|, where the sequences of sets A(n), B(n) and C(n) are introduced in A384479 and D(n) = B(n) U C(n).

Alternatively a(n) is the number of distinct five-triplet sets of the terms produced by D(n)-A(n); that is, a(n) = |D(n)-A(n)|, where A(n), B(n) and C(n) are introduced in A384479 and D(n) = B(n) U C(n).

Developed as the three dimensional extension of the Mondrian Art Problem.

Alternatively, a(n) is the optimal solution when an n X n X n cube is partitioning into 3 cuboids of different dimensions.

Let elements of the unordered integer triplet (x,y,z) be the dimensions of cuboid in a set of three cuboids.

Let V(x,y,z) = x*y*z be the volume and for a given set of triplets S, Min(S) = min{V(x,y,z):(x,y,z) in S}, Max(S) = max{V(x,y,z):(x,y,z) in S}, and defect(S) = Max(S)-Min(S).

a(n) is the least possible value of the defect as S runs over the possible partitions of the n X n X n cuboid into 3 cuboids of different dimensions.

The partition here must be valid packing of the n X n X n cube, hence T(n,k) is generally less than the number of partitions of n^3 into distinct cuboids (x,y,z) with 1 <= x,y,z <= n and volume x*y*z excluding x=y=z.