A129668
Number of different ways to divide an n X n X n cube into subcubes, considering only the list of parts.
Original entry on oeis.org
1, 2, 3, 11, 19, 121, 291, 1656
Offset: 1
a(3) = 3 because the 3 X 3 X 3 cube can be divided into subcubes in 3 different ways: a single 3 X 3 X 3 cube, a 2 X 2 X 2 plus 19 1 X 1 X 1 cubes, or 27 1 X 1 X 1 cubes.
a(4) = 11 because the 4 X 4 X 4 cube can be divided into 11 different combinations of subcubes. The table below lists each of the 11 combinations and gives the number of ways those subcubes can be arranged:
(1) 64 1 X 1 X 1 cubes in 1 way
(2) 56 1 X 1 X 1 cubes and 1 2 X 2 X 2 cube in 27 ways
(3) 48 1 X 1 X 1 cubes and 2 2 X 2 X 2 cubes in 193 ways
(4) 40 1 X 1 X 1 cubes and 3 2 X 2 X 2 cubes in 544 ways
(5) 32 1 X 1 X 1 cubes and 4 2 X 2 X 2 cubes in 707 ways
(6) 24 1 X 1 X 1 cubes and 5 2 X 2 X 2 cubes in 454 ways
(7) 16 1 X 1 X 1 cubes and 6 2 X 2 X 2 cubes in 142 ways
(8) 8 1 X 1 X 1 cubes and 7 2 X 2 X 2 cubes in 20 ways
(9) 8 2 X 2 X 2 cubes in 1 way
(10) 37 1 X 1 X 1 cubes and 1 3 X 3 X 3 cube in 8 ways
(11) 1 4 X 4 X 4 cube in 1 way
The total number of arrangements is 2098 = A228267(4,4,4).
Cf.
A034295 (same problem in 2 dimensions rather than 3).
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.
Original entry on oeis.org
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
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 ...
Showing 1-2 of 2 results.
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