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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.

%I A187800 #26 Nov 01 2021 03:20:21
%S A187800 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,
%T A187800 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,
%U A187800 142,20,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1
%N 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.
%C A187800 Row lengths are specified in A228726.
%H A187800 Christopher Hunt Gribble, <a href="/A187800/b187800.txt">Rows 1..34 flattened</a>
%H A187800 Christopher Hunt Gribble, <a href="/A187800/a187800.cpp.txt">C++ program</a>
%e A187800 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:
%e A187800 ._______.    ._______.    ._______.
%e A187800 |   |   |    | . | . |    |   |   |
%e A187800 |___|___|    |___|___|    |___|___|
%e A187800 |_|_|_|_|    |_|_|_|_|    |_|_|_|_|
%e A187800 ._______.    ._______.    ._______.
%e A187800 |   |_|_|    | . |_|_|    |   |_|_|
%e A187800 |___|   |    |___| . |    |___|   |
%e A187800 |_|_|___|    |_|_|___|    |_|_|___|
%e A187800 ._______.    ._______.    ._______.
%e A187800 |_|_|   |    |_|_| . |    |_|_|   |
%e A187800 |   |___|    | . |___|    |   |___|
%e A187800 |___|_|_|    |___|_|_|    |___|_|_|
%e A187800 ._______.    ._______.    ._______.
%e A187800 |_|_|_|_|    |_|_|_|_|    |_|_|_|_|
%e A187800 |   |   |    | . | . |    |   |   |
%e A187800 |___|___|    |___|___|    |___|___|
%e A187800 .
%e A187800 The irregular triangle begins:
%e A187800       u 0   1   2   3   4   5   6   7   8   9  10  11  12 ...
%e A187800 n k r
%e A187800 1,1,1   1
%e A187800 2,1,1   1
%e A187800 2,2,1   1
%e A187800 2,2,2   1   1
%e A187800 3,1,1   1
%e A187800 3,2,1   1
%e A187800 3,2,2   1   2
%e A187800 3,3,1   1
%e A187800 3,3,2   1   4
%e A187800 3,3,3   1   8   0   0   0   0   0   0   1
%e A187800 4,1,1   1
%e A187800 4,2,1   1
%e A187800 4,2,2   1   3   1
%e A187800 4,3,1   1
%e A187800 4,3,2   1   6   4
%e A187800 4,3,3   1  12  16   0   0   0   0   0   2
%e A187800 4,4,1   1
%e A187800 4,4,2   1   9  16   8   1
%e A187800 4,4,3   1  18  64  64  16   0   0   0   4
%e A187800 4,4,4   1  27 193 544 707 454 142  20   9   0   0   0   0 ...
%Y A187800 Row sums = A228267(n,k,r).
%Y A187800 Cf. A225777.
%K A187800 nonn,tabf
%O A187800 1,9
%A A187800 _Christopher Hunt Gribble_, Aug 30 2013