A228202 Number T(n,k,r) of partitions of an n X k X r rectangular cuboid into integer-sided cubes, considering only the list of parts; irregular triangle T(n,k,r), n >= k >= r >= 1 read by rows.

1, 1, 1, 2, 1, 1, 2, 1, 2, 3, 1, 1, 3, 1, 3, 4, 1, 5, 6, 11, 1, 1, 3, 1, 3, 5, 1, 5, 8, 15, 1, 5, 9, 17, 19, 1, 1, 4, 1, 4, 7, 1, 7, 11, 24, 1, 7, 16, 34, 40, 1, 10, 23, 52, 80, 121

Offset

1,4

Links

Alois P. Heinz, Rows n = 1..6, flattened (rows n=1..5 from Christopher Hunt Gribble)

Christopher Hunt Gribble, C++ program

Example

The irregular triangle begins:
   r  1   2   3   4 ...
n,k
1,1   1
2,1   1
2,2   1   2
3,1   1
3,2   1   2
3,3   1   2   3
4,1   1
4,2   1   3
4,3   1   3   4
4,4   1   5   6  11
5,1   1
5,2   1   3
5,3   1   3   5
5,4   1   5   8  15
5,5   1   5   9  17  19
...
T(4,4,3) = 6 because there are 6 partitions of a 4 X 4 X 3 rectangular cuboid into integer-sided cubes.  The partitions are:
48 1 X 1 X 1 cubes,
40 1 X 1 X 1 cubes and 1 2 X 2 X 2 cube,
32 1 X 1 X 1 cubes and 2 2 X 2 X 2 cubes,
24 1 X 1 X 1 cubes and 3 2 X 2 X 2 cubes,
16 1 X 1 X 1 cubes and 4 2 X 2 X 2 cubes,
21 1 X 1 X 1 cubes and 1 3 X 3 X 3 cube.

Crossrefs

T(n,n,n) = A129668(n).

Cf. A224697.

Sequence in context: A317685 A257540 A333516 * A308972 A137900 A025837

Adjacent sequences: A228199 A228200 A228201 * A228203 A228204 A228205

Keyword

nonn,tabf

Author

Christopher Hunt Gribble, Aug 17 2013

Extensions

21 more terms (row 6) from Alois P. Heinz, Aug 18 2013

Status

approved