A187800 Number T(n,k,r,u) of dissections of an n X k X r rectangular cuboid on a unit cubic grid into integer-sided cubes containing u nodes that are unconnected to any of their neighbors; irregular triangle T(n,k,r,u), n >= k >= r >= 1, u >= 0 read by rows.
1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 4, 1, 8, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 3, 1, 1, 1, 6, 4, 1, 12, 16, 0, 0, 0, 0, 0, 2, 1, 1, 9, 16, 8, 1, 1, 18, 64, 64, 16, 0, 0, 0, 4, 1, 27, 193, 544, 707, 454, 142, 20, 9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1
Offset: 1
Examples
T(4,3,2,2) = 4 because the 4 X 3 X 2 rectangular cuboid can be dissected in 4 distinct ways in which there are 2 nodes unconnected to any of their neighbors. The dissections 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 ...
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 2
3,3,1 1
3,3,2 1 4
3,3,3 1 8 0 0 0 0 0 0 1
4,1,1 1
4,2,1 1
4,2,2 1 3 1
4,3,1 1
4,3,2 1 6 4
4,3,3 1 12 16 0 0 0 0 0 2
4,4,1 1
4,4,2 1 9 16 8 1
4,4,3 1 18 64 64 16 0 0 0 4
4,4,4 1 27 193 544 707 454 142 20 9 0 0 0 0 ...
Links
- Christopher Hunt Gribble, Rows 1..34 flattened
- Christopher Hunt Gribble, C++ program
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