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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. 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,63

COMMENTS

Row lengths are specified in A228726.

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

EXAMPLE

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:

._______.    ._______.    ._______.    ._______.    ._______.

|   |   |    | . | . |    |   |   |    | . | . |    |   |   |

|___|___|    |___|___|    |___|___|    |___|___|    |___|___|

|   |   |    | . | . |    |   |   |    | . | . |    |   |   |

|___|___|    |___|___|    |___|___|    |___|___|    |___|___|

._______.    ._______.    ._______.    ._______.    ._______.

|     |_|    | . . |_|    | . . |_|    |     |_|    |_|_|_|_|

|     |_|    | . . |_|    | . . |_|    |     |_|    |_|_|_|_|

|_____|_|    |_____|_|    |_____|_|    |_____|_|    |_|_|_|_|

|_|_|_|_|    |_|_|_|_|    |_|_|_|_|    |_|_|_|_|    |_|_|_|_|

CROSSREFS

Row sums = A228202(n,k,r)

Cf. A225542

Sequence in context: A130779 A130706 A000038 * A281669 A014083 A238403

Adjacent sequences:  A228591 A228592 A228593 * A228595 A228596 A228597

KEYWORD

nonn,tabf

AUTHOR

Christopher Hunt Gribble, Aug 27 2013

STATUS

approved

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Last modified December 13 03:41 EST 2019. Contains 329968 sequences. (Running on oeis4.)