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.

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

Author

Christopher Hunt Gribble, Aug 19 2013

Keywords

• nonn
• tabf

Comments

The main diagonal T(n,n,n) is 1, 2, 10, 2098, 4006722, .... - R. J. Mathar and Rob Pratt, Nov 27 2017

Examples

			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:
  ._____.    ._____.    ._____.
  |_|_|_|    |_|_|_|    |_|_|_|
  |_|_|_|    |_|_|_|    |_|_|_|
  ._____.    ._____.    ._____.
  |   |_|    |   |_|    |   |_|
  |___|_|    |___|_|    |___|_|
  ._____.    ._____.    ._____.
  |_|   |    |_|   |    |_|   |
  |_|___|    |_|___|    |_|___|
		

Links

• Christopher Hunt Gribble, C++ program

Crossrefs

Cf. A219924.

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

Extensions

20 more terms from R. J. Mathar, Dec 03 2017