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A216274 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. 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 (list; table; graph; refs; listen; history; text; internal format)
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+dim-1, {dim}], 1] &, maxHeight], 1]; pairs = lexicographicLattice[{2, 12}]-1; Table[getvalue[First[pairs[[j]]], Last[pairs[[j]]]+1], {j, 1, Length[pairs]}]
CROSSREFS
Sequence in context: A163491 A080772 A259324 * A145111 A104795 A347570
KEYWORD
nonn,tabl
AUTHOR
Frank M Jackson, Mar 16 2013
EXTENSIONS
Edited by N. J. A. Sloane, May 20 2023
STATUS
approved

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Last modified February 27 08:11 EST 2024. Contains 370367 sequences. (Running on oeis4.)