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

1,9

Comments

Row lengths are specified in A228726.

Links

Christopher Hunt Gribble, Rows 1..34 flattened

Christopher Hunt Gribble, C++ program

Example

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 ...

Crossrefs

Row sums = A228267(n,k,r).

Cf. A225777.

Sequence in context: A375023 A021477 A124939 * A340189 A323873 A365582

Adjacent sequences: A187797 A187798 A187799 * A187801 A187802 A187803

Keyword

nonn,tabf

Author

Christopher Hunt Gribble, Aug 30 2013

Status

approved