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A384479 a(n) is the number of ways to partition n X n X n cube into 5 noncongruent cuboids.

%I A384479 #43 Aug 04 2025 04:13:21
%S A384479 0,0,2,31,209,560,1561,2852,5894,9093,15778,22304,34805,46194,67399,
%T A384479 85610,118461,146218,194363,233941,302115,356523,448831,522153,643781,
%U A384479 738976,896563,1017619,1216376,1368946,1615516,1803028,2106173,2333683,2699501,2974256,3410946
%N A384479 a(n) is the number of ways to partition n X n X n cube into 5 noncongruent cuboids.
%C A384479 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.
%C A384479 First stage:
%C A384479 Generating the sequence of sets A(n) of five triplets of distinct element composition by decomposing the triplet (n,n,n).
%C A384479 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)}}.
%C A384479 Therefore the number of sets in A(3) is |A(3)| = 2.
%C A384479 Second stage:
%C A384479 Generating the sequence of sets B(n) of five triplets of distinct element composition by decomposing the triplet (n,n,n).
%C A384479 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.
%C A384479 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.
%C A384479 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.
%C A384479 Third stage:
%C A384479 Generating the sequence of sets C(n) of five triplets of distinct element composition by decomposing the triplet (n,n,n).
%C A384479 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.
%C A384479 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.
%C A384479 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.
%C A384479 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)|.
%H A384479 Janaka Rodrigo, <a href="/A384479/a384479_3.pdf">Sets in Triplets Form and Two Dimensional Illustrations for Partitioning</a>
%e A384479 a(1) = a(2) = 0, since the cubes 1 X 1 X 1 and 2 X 2 X 2 can never be partitioned into five cuboids.
%e A384479 a(3) = |A(3)| + |B(3)| + |C(3)| - |A(3) intersection B(3)| - |A(3) intersection C(3)|= 2 + 1 + 0 - 1 - 0 = 2.
%e A384479 a(4) = |A(4)| + |B(4)| + |C(4)| - |A(4) intersection B(4)| - |A(4) intersection C(4)|= 30 + 3 + 1 - 2 - 1 = 31.
%Y A384479 Cf. A381847, A384208, A384311, A384511, A385247, A386296.
%K A384479 nonn
%O A384479 1,3
%A A384479 _Janaka Rodrigo_, May 30 2025
%E A384479 a(11)-a(16) from _Sean A. Irvine_, Jul 26 2025
%E A384479 More terms from _Jinyuan Wang_, Aug 03 2025