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T(n,k,r) is the total number of parts in the set of partitions of an n X k X r rectangular cuboid into integer-sided cubes, considering only the list of parts; irregular triangle T(n,k,r), n >= k >= r >= 1, read by rows.
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%I #22 Sep 06 2021 04:25:08

%S 1,2,4,9,3,6,17,9,29,48,4,8,27,12,51,97,16,90,192,363,5,10,39,15,69,

%T 145,20,130,311,685,25,180,459,1056

%N T(n,k,r) is the total number of parts in the set of partitions of an n X k X r rectangular cuboid into integer-sided cubes, considering only the list of parts; irregular triangle T(n,k,r), n >= k >= r >= 1, read by rows.

%H Christopher Hunt Gribble, <a href="/A227272/a227272.cpp.txt">C++ program</a>

%F T(n,k,1) = n*k.

%F T(n,2,2) = (n+1)(n+2) - 3.

%e The irregular triangle begins:

%e . r 1 2 3 4 ...

%e n,k

%e 1,1 1

%e 2,1 2

%e 2,2 4 9

%e 3,1 3

%e 3,2 6 17

%e 3,3 9 29 48

%e 4,1 4

%e 4,2 8 27

%e 4,3 12 51 97

%e 4,4 16 90 192 363

%e 5,1 5

%e 5,2 10 39

%e 5,3 15 69 145

%e 5,4 20 130 311 685

%e 5,5 25 180 459 1056 ...

%e ...

%e T(2,2,2) = 9 because a 2 X 2 X 2 rectangular cuboid has 2 partitions, (8 1 X 1 X 1 squares) and (1 2 X 2 X 2 square) with 9 parts in total.

%Y Cf. A225622.

%K nonn,tabf

%O 1,2

%A _Christopher Hunt Gribble_, Sep 02 2013