A228594 Triangle T(n,k,r,u) read by rows: number of partitions of an n X k X r rectangular cuboid on a cubic grid into integer-sided cubes containing u nodes that are unconnected to any of their neighbors, considering only the number of parts; irregular triangle T(n,k,r,u), n >= k >= r >= 1, u >= 0.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Author

Christopher Hunt Gribble, Aug 27 2013

Keywords

• nonn
• tabf

Comments

Row lengths are specified in A228726.

Examples

			T(4,4,4,8) = 2 because the 4 X 4 X 4 rectangular cuboid (in this case a cube) has 2 partitions in which there are 8 nodes unconnected to any of their neighbors.  The partitions are (8 2 X 2 X 2 cubes) and (37 1 X 1 X 1 cubes and 1 3 X 3 X 3 cube).  The partitions and isolated nodes can be illustrated by expanding into 2 dimensions:
._______.    ._______.    ._______.    ._______.    ._______.
|   |   |    | . | . |    |   |   |    | . | . |    |   |   |
|___|___|    |___|___|    |___|___|    |___|___|    |___|___|
|   |   |    | . | . |    |   |   |    | . | . |    |   |   |
|___|___|    |___|___|    |___|___|    |___|___|    |___|___|
._______.    ._______.    ._______.    ._______.    ._______.
|     |_|    | . . |_|    | . . |_|    |     |_|    |_|_|_|_|
|     |_|    | . . |_|    | . . |_|    |     |_|    |_|_|_|_|
|_____|_|    |_____|_|    |_____|_|    |_____|_|    |_|_|_|_|
|_|_|_|_|    |_|_|_|_|    |_|_|_|_|    |_|_|_|_|    |_|_|_|_|
.
The irregular triangle begins:
      u 0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 ...
n k r
1,1,1   1
2,1,1   1
2,2,1   1
2,2,2   1  1
3,1,1   1
3,2,1   1
3,2,2   1  1
3,3,1   1
3,3,2   1  1
3,3,3   1  1  0  0  0  0  0  0  1
4,1,1   1
4,2,1   1
4,2,2   1  1  1
4,3,1   1
4,3,2   1  1  1
4,3,3   1  1  1  0  0  0  0  0  1
4,4,1   1
4,4,2   1  1  1  1  1
4,4,3   1  1  1  1  1  0  0  0  1
4,4,4   1  1  1  1  1  1  1  1  2  0  0  0  0  0  0  0  0 ...
5,1,1   1
5,2,1   1
5,2,2   1  1  1
5,3,1   1
5,3,2   1  1  1
5,3,3   1  1  1  0  0  0  0  0  1  1
5,4,1   1
5,4,2   1  1  1  1  1
5,4,3   1  1  1  1  1  0  0  0  1  1  1
5,4,4   1  1  1  1  1  1  1  1  2  1  1  1  1  0  0  0  0 ...
5,5,1   1
5,5,2   1  1  1  1  1
5,5,3   1  1  1  1  1  0  0  0  1  1  1  1
5,5,4   1  1  1  1  1  1  1  1  2  1  1  1  1  1  1  0  0 ...
		

Links

• Christopher Hunt Gribble, Rows 1..34 flattened
• Christopher Hunt Gribble, C++ program

Crossrefs

Row sums = A228202(n,k,r).

Cf. A225542.