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

Author

Christopher Hunt Gribble, Aug 17 2013

Keywords

• nonn
• tabf

Examples

			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.
		

Links

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

Crossrefs

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

Cf. A224697.

Extensions

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