A386847 -id:A386847 - cp's OEIS frontend

A386848 Array read by descending antidiagonals:T(n,k) is the number of ways to partition n X n X n cube into k noncongruent cuboids excluding strict cuboids.

1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 2, 0, 2, 1, 0, 0, 1, 3, 0, 2, 1, 0, 0, 1, 3, 4, 1, 3, 1, 0, 0, 0, 1, 10, 6, 1, 3, 1, 0, 0, 0, 6, 9, 19, 6, 2, 4, 1, 0, 0, 0, 5, 34, 24, 30, 9, 3, 4, 1, 0, 0, 0, 0, 78, 37, 47, 44, 8, 4, 5, 1, 0, 0, 0, 0, 93

Offset: 1

Janaka Rodrigo, Aug 05 2025

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

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 excluding x != y != z.

			1    0    0    0    0
1    0    0    0    0
1    1    0    2    1
1    1    0    3    3
1    2    0    4   10
1    2    1    6   19
1    3    1    6   30
1    3    2    9   44
1    4    3    8   64
1    4    4    13  84

Sean A. Irvine, Table of n, a(n) for n = 1..140

Cf. A386296.

Columns: A004526(k=2), A211540(k=3), A386846(k=4), A386847(k=5).

T(n,1) = 1,

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

More terms from Sean A. Irvine, Aug 05 2025

A386902 a(n) is the number of distinct five-cuboid combinations that fill an n X n X n with only strict cuboids.

Original entry on oeis.org

0, 0, 0, 0, 0, 18, 74, 193, 491, 857, 1695, 2503, 4321, 5836, 9200, 11715, 17284, 21256, 29805, 35589, 48156, 56260, 73766, 84860, 108495, 123080, 154298, 172998, 213045, 236895, 287260, 316743, 379465, 415456, 491930, 535713, 627879, 680052, 790401, 851914, 982130

Offset: 1

Janaka Rodrigo, Aug 07 2025

A strict cuboid is a cuboid with all three dimensions different.

Alternatively a(n) is the number of ways to decompose (n,n,n) triplet into set of geometrically feasible distinct five unordered triplets of the form (x,y,z) with x != y != z for each of five triplets.

			(6,6,6) triplet can be decomposed into set of five triplets in 560 different ways and only 18 of those formed by only strict cuboids. Three of those sets are given below:
   {(1,2,3), (1,3,4), (2,3,6), (3,4,6), (3,5,6)},
   {(1,2,6), (1,4,6), (2,4,6), (2,5,6), (3,4,6)},
   {(1,3,4), (1,3,6), (2,3,5), (2,3,6), (4,5,6)}.

Cf. A384479, A386847.

a(16)-a(18) from Sean A. Irvine, Aug 14 2025

More terms from Jinyuan Wang, Aug 29 2025

cp's OEIS Frontend

A386848 Array read by descending antidiagonals:T(n,k) is the number of ways to partition n X n X n cube into k noncongruent cuboids excluding strict cuboids.

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