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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.

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.

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.

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 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.

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 five cuboids of different dimensions.

Let elements of the unordered integer triplet (x,y,z) be the dimensions of a cuboid in a set of five cuboids and volume V(x,y,z) = x*y*z; cuboids have five values for each set of five triplets S, produced by the union of A(n), B(n), C(n), where A(n), B(n), and C(n) are sequences of sets as introduced in A384479.

Define min(S) = min{V(x,y,z):(x,y,z) in S} and max(S) = max{V(x,y,z):(x,y,z) in S}, then defect(S) = max(S) - min(S).

a(n) is the smallest possible value of defect(S) where S runs over all possible ways of partitioning the n X n X n cube into five cuboids of different dimensions.

A strict cuboid is a cuboid with all three dimensions different to each other.

Alternatively a(n) is number of ways to decompose (n,n,n) triplet into sets of distinct unordered geometrically feasible five triplets of the form (x,y,z) excluding x != y != z in any of the triplets.

Alternatively a(n) is the number of ways to decompose (n,n,n) triplet into sets of distinct five unordered triplets of the form (x,y,z) without having x = y = z in any of the triplets.