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

1,18

Comments

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.

Links

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

Formula

T(n,1) = 1,

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

Example

Array begins:
  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   ...
  ...

Crossrefs

Cf. A386296.

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

Sequence in context: A006996 A321430 A343914 * A262774 A112604 A203399

Adjacent sequences: A386845 A386846 A386847 * A386849 A386850 A386851

Keyword

nonn,tabl

Author

Janaka Rodrigo, Aug 05 2025

Extensions

More terms from Sean A. Irvine, Aug 05 2025

Status

approved