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A228267
Number T(n,k,r) of dissections of an n X k X r rectangular cuboid into integer-sided cubes including rotations and reflections; irregular triangle T(n,k,r), n >= k >= r >= 1 read by rows.
3
1, 1, 1, 2, 1, 1, 3, 1, 5, 10, 1, 1, 5, 1, 11, 31, 1, 35, 167, 2098, 1, 1, 8, 1, 21, 76, 1, 93, 635, 15511, 1, 314, 3354, 185473, 4006722, 1, 1, 13, 1, 43, 210, 1, 269, 2887, 151378, 1, 1213, 22478, 3243515, 143662050, 1, 6427, 235150, 112411358
OFFSET
1,4
COMMENTS
The main diagonal T(n,n,n) is 1, 2, 10, 2098, 4006722, .... - R. J. Mathar and Rob Pratt, Nov 27 2017
LINKS
Christopher Hunt Gribble, C++ program
FORMULA
T(1,1,r) = T(n,n,1) = 1. - R. J. Mathar, Dec 03 2017
T(2,2,r) = A000045(r+1). - R. J. Mathar, Dec 03 2017
T(3,3,r>=1) = 1, 5, 10, 31, ... with g.f. 1/(1-x-4*x^2-x^3). - R. J. Mathar, Dec 03 2017
T(4,4,r>=1) = 1, 35, 167, 2098, 15511, 151378, 1272179, 11574563, 100928230, 900224006, ... with TBD rational g.f. - R. J. Mathar, Dec 03 2017
T(n,n,2) = A063443(n). - R. J. Mathar, Dec 03 2017
EXAMPLE
The irregular triangle begins:
. r 1 2 3 4 ...
n,k
1,1 1
2,1 1
2,2 1 2
3,1 1
3,2 1 3
3,3 1 5 10
4,1 1
4,2 1 5
4,3 1 11 31
4,4 1 35 167 2098
5,1 1
5,2 1 8
5,3 1 21 76
5,4 1 93 635 15511
5,5 1 314 3354 185473 ...
...
T(3,2,2) = 3 because there are 3 distinct dissections of a 3 X 2 X 2 rectangular cuboid into integer-sided cubes. The dissections expanded into 2 dimensions are:
._____. ._____. ._____.
|_|_|_| |_|_|_| |_|_|_|
|_|_|_| |_|_|_| |_|_|_|
._____. ._____. ._____.
| |_| | |_| | |_|
|___|_| |___|_| |___|_|
._____. ._____. ._____.
|_| | |_| | |_| |
|_|___| |_|___| |_|___|
CROSSREFS
Cf. A219924.
Sequence in context: A327981 A348447 A277606 * A170820 A339615 A003687
KEYWORD
nonn,tabf
AUTHOR
EXTENSIONS
20 more terms from R. J. Mathar, Dec 03 2017
STATUS
approved