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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.

%I A228267 #25 Sep 06 2021 08:30:01
%S A228267 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,
%T A228267 635,15511,1,314,3354,185473,4006722,1,1,13,1,43,210,1,269,2887,
%U A228267 151378,1,1213,22478,3243515,143662050,1,6427,235150,112411358
%N 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.
%C A228267 The main diagonal T(n,n,n) is 1, 2, 10, 2098, 4006722, .... - _R. J. Mathar_ and _Rob Pratt_, Nov 27 2017
%H A228267 Christopher Hunt Gribble, <a href="/A228267/a228267.cpp.txt">C++ program</a>
%F A228267 T(1,1,r) = T(n,n,1) = 1. - _R. J. Mathar_, Dec 03 2017
%F A228267 T(2,2,r) = A000045(r+1). - _R. J. Mathar_, Dec 03 2017
%F A228267 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
%F A228267 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
%F A228267 T(n,n,2) = A063443(n). - _R. J. Mathar_, Dec 03 2017
%e A228267 The irregular triangle begins:
%e A228267 .   r 1      2      3      4 ...
%e A228267 n,k
%e A228267 1,1   1
%e A228267 2,1   1
%e A228267 2,2   1      2
%e A228267 3,1   1
%e A228267 3,2   1      3
%e A228267 3,3   1      5     10
%e A228267 4,1   1
%e A228267 4,2   1      5
%e A228267 4,3   1     11     31
%e A228267 4,4   1     35    167   2098
%e A228267 5,1   1
%e A228267 5,2   1      8
%e A228267 5,3   1     21     76
%e A228267 5,4   1     93    635  15511
%e A228267 5,5   1    314   3354 185473 ...
%e A228267 ...
%e A228267 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:
%e A228267   ._____.    ._____.    ._____.
%e A228267   |_|_|_|    |_|_|_|    |_|_|_|
%e A228267   |_|_|_|    |_|_|_|    |_|_|_|
%e A228267   ._____.    ._____.    ._____.
%e A228267   |   |_|    |   |_|    |   |_|
%e A228267   |___|_|    |___|_|    |___|_|
%e A228267   ._____.    ._____.    ._____.
%e A228267   |_|   |    |_|   |    |_|   |
%e A228267   |_|___|    |_|___|    |_|___|
%Y A228267 Cf. A219924.
%K A228267 nonn,tabf
%O A228267 1,4
%A A228267 _Christopher Hunt Gribble_, Aug 19 2013
%E A228267 20 more terms from _R. J. Mathar_, Dec 03 2017