A386903 Array read by descending antidiagonals: T(n,k) is the number of ways to partition n X n X n cube into k noncongruent strict cuboids, n>=5, k>=4.

1, 0, 2, 3, 18, 9, 1, 64, 74, 12, 1, 143, 450, 193, 30, 0, 197, 2090, 1769, 491, 36, 0, 156, 8039, 13441, 5687, 857, 70, 0, 57, 24641, 88001, 56540, 12994, 1695, 80, 0, 5

Offset: 5

Author

Janaka Rodrigo, Aug 07 2025

Keywords

• tabl
• nonn
• more

Comments

A strict cuboid is a cuboid with all dimensions different.

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, x != y != z and volume x*y*z.

There are no solutions for n < 5 or k < 4.

Examples

			Array begins:
  n\k|  4     5      6       7      8      9
  ---+--------------------------------------
   5 |  1     0      3       1      1      0
   6 |  2    18     64     143    197    156
   7 |  9    74    450    2090   8039  24641
   8 | 12   193   1769   13441  88001      ?
   9 | 30   491   5687   56540      ?      ?
  10 | 36   857  12994  170052      ?      ?
  ...
		

Crossrefs

Cf. A386296.

Columns: A386884 (k=4), A386902 (k=5).