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
KEYWORD
nonn,tabl
AUTHOR
Frank M Jackson, Mar 16 2013
EXTENSIONS
Edited by N. J. A. Sloane, May 20 2023
STATUS
approved