A385580 -id:A385580 - cp's OEIS frontend

A387040 a(n) is the number of distinct five-cuboid combinations that fill an n X n X n cube with cuboids of different volumes.

0, 0, 2, 26, 206, 442, 1531, 2661, 5574, 8514, 15614, 20331, 34500, 44814, 64503, 83143, 117759, 141290, 193436, 226722, 295978, 351953, 447208, 507508, 637447, 732322, 887044, 1001577, 1213233, 1337525, 1611692, 1786560, 2088648, 2321052, 2673275, 2929254, 3404667

Offset: 1

Janaka Rodrigo, Aug 14 2025

Alternatively a(n) is the number of ways to decompose (n,n,n) triplet into geometrically feasible five distinct unordered triplets of the form (x,y,z) with no pair having equal value for the product x*y*z.

			According to A384479(5), (5,5,5) triplet can be decomposed into 209 distinct sets of five triplets and only three of them contain pair of triplets with equal value for x*y*z. Those are,
   {(1,2,5), (1,3,5), (1,4,5), (2,2,5), (3,4,5)},
   {(1,1,5), (1,4,5), (2,2,5), (2,3,5), (2,5,5)},
   {(1,3,5), (1,4,5), (2,2,5), (2,3,5), (2,4,5)}.
Therefore a(5) = 209-3 = 206.

Cf. A384479, A385580, A387121.

a(15)-a(16) from Sean A. Irvine, Aug 19 2025

More terms from Jinyuan Wang, Aug 29 2025

A387121 Array read by antidiagonals: T(n,k) is the number of sets of k integer sided cuboids with distinct volumes that fill an n X n X n cube.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 2, 1, 1, 0, 0, 4, 3, 2, 1, 0, 0, 2, 11, 8, 2, 1, 0, 0, 1, 26, 47, 11, 3, 1, 0, 0, 0, 55, 206, 77, 19, 3, 1, 0, 0, 0, 48, 793, 442, 183, 23, 4, 1, 0, 0, 0, 23, 2653, 2451, 1531, 259, 35, 4, 1, 0, 0, 0, 0, 6706, 13022, 12178

Offset: 1

Janaka Rodrigo, Aug 16 2025

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 no pair of triplets having equal volume x*y*z.

			Array begins:
  1     0     0     0     0
  1     0     0     0     0
  1     1     2     4     2
  1     1     3    11    26
  1     2     8    47   206
  1     2    11    77   442
  1     3    19   183  1531
  1     3    23   259  2661
  1     4    35   457  5574
  1     4    40   599  8514
  ...

Columns are A004526 (k=2), A381847 (k=3), A385580 (k=4), A387040 (k=5).

T(n,1) = 1, T(n,k) = 0 for k > n^3.

More terms from Sean A. Irvine, Aug 25 2025

cp's OEIS Frontend

A387040 a(n) is the number of distinct five-cuboid combinations that fill an n X n X n cube with cuboids of different volumes.

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