

A216274


Square array A(n,k) = maximal number of regions into which kspace can be divided by n hyperplanes (k >= 1, n >= 0), read by antidiagonals.


1



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, 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, 16, 32, 63, 99, 93, 46, 11, 1, 2, 4, 8, 16, 32, 64, 120, 163, 130, 56, 12
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OFFSET

0,3


COMMENTS

For all fixed k, the sequences A(n,k) are "complete" (sic).
This array is similar to A145111 with first variation at 34th term.


LINKS

Wikipedia, "Complete" sequence. [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]


FORMULA

A(k,n) = Sum_{i=0..k} C(n, i), k >=1, n >= 0.


EXAMPLE

Square array A(n,k) begins:
1, 1, 1, 1, 1, 1, ...
2, 2, 2, 2, 2, 2, ...
3, 4, 4, 4, 4, 4, ...
4, 7, 8, 8, 8, 8, ...
5, 11, 15, 16, 16, 16, ...
6, 16, 26, 31, 32, 32, ...
So the maximal number of pieces into which a cube can be divided after 5 planar cuts is A(5,3) = 26.


MATHEMATICA

getvalue[n_, k_] := Sum[Binomial[n, i], {i, 0, k}]; lexicographicLattice[{dim_, maxHeight_}] := Flatten[Array[Sort@Flatten[(Permutations[#1] &) /@IntegerPartitions[#1+dim1, {dim}], 1] &, maxHeight], 1]; pairs = lexicographicLattice[{2, 12}]1; Table[getvalue[First[pairs[[j]]], Last[pairs[[j]]]+1], {j, 1, Length[pairs]}]


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STATUS

approved



