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%I #28 Jun 12 2023 08:47:44
%S 1,1,2,1,2,3,1,2,4,4,1,2,4,7,5,1,2,4,8,11,6,1,2,4,8,15,16,7,1,2,4,8,
%T 16,26,22,8,1,2,4,8,16,31,42,29,9,1,2,4,8,16,32,57,64,37,10,1,2,4,8,
%U 16,32,63,99,93,46,11,1,2,4,8,16,32,64,120,163,130,56,12
%N Square array A(n,k) = maximal number of regions into which k-space can be divided by n hyperplanes (k >= 1, n >= 0), read by antidiagonals.
%C For all fixed k, the sequences A(n,k) are "complete" (sic).
%C This array is similar to A145111 with first variation at 34th term.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Complete_sequence">"Complete" sequence</a>. [Wikipedia calls a sequence "complete" (sic) if every positive integer is a sum of distinct terms. This name is extremely misleading and should be avoided. - _N. J. A. Sloane_, May 20 2023]
%F A(k,n) = Sum_{i=0..k} C(n, i), k >=1, n >= 0.
%e Square array A(n,k) begins:
%e 1, 1, 1, 1, 1, 1, ...
%e 2, 2, 2, 2, 2, 2, ...
%e 3, 4, 4, 4, 4, 4, ...
%e 4, 7, 8, 8, 8, 8, ...
%e 5, 11, 15, 16, 16, 16, ...
%e 6, 16, 26, 31, 32, 32, ...
%e So the maximal number of pieces into which a cube can be divided after 5 planar cuts is A(5,3) = 26.
%t getvalue[n_, k_] := Sum[Binomial[n, i], {i, 0, k}]; lexicographicLattice[{dim_, maxHeight_}] := Flatten[Array[Sort@Flatten[(Permutations[#1] &) /@IntegerPartitions[#1+dim-1, {dim}], 1] &, maxHeight], 1]; pairs = lexicographicLattice[{2, 12}]-1; Table[getvalue[First[pairs[[j]]], Last[pairs[[j]]]+1], {j, 1, Length[pairs]}]
%Y Cf. A000124, A000125, A059214.
%K nonn,tabl
%O 0,3
%A _Frank M Jackson_, Mar 16 2013
%E Edited by _N. J. A. Sloane_, May 20 2023
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