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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.

%I A228594 #20 Oct 31 2021 07:46:29
%S A228594 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,
%T A228594 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,
%U A228594 0,0,0,0,0,0,0,0,0,0,0,0,0,1
%N 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.
%C A228594 Row lengths are specified in A228726.
%H A228594 Christopher Hunt Gribble, <a href="/A228594/b228594.txt">Rows 1..34 flattened</a>
%H A228594 Christopher Hunt Gribble, <a href="/A228594/a228594.cpp.txt">C++ program</a>
%e A228594 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:
%e A228594 ._______.    ._______.    ._______.    ._______.    ._______.
%e A228594 |   |   |    | . | . |    |   |   |    | . | . |    |   |   |
%e A228594 |___|___|    |___|___|    |___|___|    |___|___|    |___|___|
%e A228594 |   |   |    | . | . |    |   |   |    | . | . |    |   |   |
%e A228594 |___|___|    |___|___|    |___|___|    |___|___|    |___|___|
%e A228594 ._______.    ._______.    ._______.    ._______.    ._______.
%e A228594 |     |_|    | . . |_|    | . . |_|    |     |_|    |_|_|_|_|
%e A228594 |     |_|    | . . |_|    | . . |_|    |     |_|    |_|_|_|_|
%e A228594 |_____|_|    |_____|_|    |_____|_|    |_____|_|    |_|_|_|_|
%e A228594 |_|_|_|_|    |_|_|_|_|    |_|_|_|_|    |_|_|_|_|    |_|_|_|_|
%e A228594 .
%e A228594 The irregular triangle begins:
%e A228594       u 0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 ...
%e A228594 n k r
%e A228594 1,1,1   1
%e A228594 2,1,1   1
%e A228594 2,2,1   1
%e A228594 2,2,2   1  1
%e A228594 3,1,1   1
%e A228594 3,2,1   1
%e A228594 3,2,2   1  1
%e A228594 3,3,1   1
%e A228594 3,3,2   1  1
%e A228594 3,3,3   1  1  0  0  0  0  0  0  1
%e A228594 4,1,1   1
%e A228594 4,2,1   1
%e A228594 4,2,2   1  1  1
%e A228594 4,3,1   1
%e A228594 4,3,2   1  1  1
%e A228594 4,3,3   1  1  1  0  0  0  0  0  1
%e A228594 4,4,1   1
%e A228594 4,4,2   1  1  1  1  1
%e A228594 4,4,3   1  1  1  1  1  0  0  0  1
%e A228594 4,4,4   1  1  1  1  1  1  1  1  2  0  0  0  0  0  0  0  0 ...
%e A228594 5,1,1   1
%e A228594 5,2,1   1
%e A228594 5,2,2   1  1  1
%e A228594 5,3,1   1
%e A228594 5,3,2   1  1  1
%e A228594 5,3,3   1  1  1  0  0  0  0  0  1  1
%e A228594 5,4,1   1
%e A228594 5,4,2   1  1  1  1  1
%e A228594 5,4,3   1  1  1  1  1  0  0  0  1  1  1
%e A228594 5,4,4   1  1  1  1  1  1  1  1  2  1  1  1  1  0  0  0  0 ...
%e A228594 5,5,1   1
%e A228594 5,5,2   1  1  1  1  1
%e A228594 5,5,3   1  1  1  1  1  0  0  0  1  1  1  1
%e A228594 5,5,4   1  1  1  1  1  1  1  1  2  1  1  1  1  1  1  0  0 ...
%Y A228594 Row sums = A228202(n,k,r).
%Y A228594 Cf. A225542.
%K A228594 nonn,tabf
%O A228594 1,63
%A A228594 _Christopher Hunt Gribble_, Aug 27 2013