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A216274
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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.
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1
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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
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0,3
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COMMENTS
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For all fixed k, the sequences A(n,k) are complete.
This array is similar to A145111 with first variation at 34th term.
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LINKS
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Table of n, a(n) for n=0..77.
Wikipedia, Complete sequence
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FORMULA
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A(k,n) = Sum_{i=0..k} C(n, i), k >=1, n >= 0.
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EXAMPLE
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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 number of maximal pieces that a cube can be divided into after 5 planar cuts is A(5,3) = 26.
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MATHEMATICA
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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]}]
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CROSSREFS
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Cf. A000124, A000125, A059214.
Sequence in context: A163491 A080772 A259324 * A145111 A104795 A347570
Adjacent sequences: A216271 A216272 A216273 * A216275 A216276 A216277
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KEYWORD
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nonn,tabl
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AUTHOR
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Frank M Jackson, Mar 16 2013
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STATUS
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approved
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