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, 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

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

Table of n, a(n) for n=0..77.

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

Cf. A000124, A000125, A059214.

Sequence in context: A163491 A080772 A259324 * A145111 A104795 A347570

Adjacent sequences: A216271 A216272 A216273 * A216275 A216276 A216277

Keyword

nonn,tabl

Author

Frank M Jackson, Mar 16 2013

Extensions

Edited by N. J. A. Sloane, May 20 2023

Status

approved