A227272 T(n,k,r) is the total number of parts in the set 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, 2, 4, 9, 3, 6, 17, 9, 29, 48, 4, 8, 27, 12, 51, 97, 16, 90, 192, 363, 5, 10, 39, 15, 69, 145, 20, 130, 311, 685, 25, 180, 459, 1056

Offset: 1

Author

Christopher Hunt Gribble, Sep 02 2013

Keywords

• nonn
• tabf

Examples

			The irregular triangle begins:
.   r   1    2    3    4 ...
n,k
1,1     1
2,1     2
2,2     4    9
3,1     3
3,2     6   17
3,3     9   29   48
4,1     4
4,2     8   27
4,3    12   51   97
4,4    16   90  192  363
5,1     5
5,2    10   39
5,3    15   69  145
5,4    20  130  311  685
5,5    25  180  459 1056 ...
...
T(2,2,2) = 9 because a 2 X 2 X 2 rectangular cuboid has 2 partitions, (8 1 X 1 X 1 squares) and (1 2 X 2 X 2 square) with 9 parts in total.
		

Links

• Christopher Hunt Gribble, C++ program

Crossrefs

Cf. A225622.

Formula

T(n,k,1) = n*k.

T(n,2,2) = (n+1)(n+2) - 3.