OFFSET
0,6
COMMENTS
A(n,k) is also the size of the Hamming ball in {0,1}^n of radius (k-1)/2 if k is odd and of the union of two Hamming balls in {0,1}^n of radius k/2-1 whose centers are of Hamming distance 1 if k is even.
LINKS
Noga Alon, Zhihan Jin, and Benny Sudakov, The Helly number of Hamming balls and related problems, arXiv:2405.10275 [math.CO], 2024. See p. 3.
S. L. Bezrukov, Specification of all maximal subsets of the unit cube with respect to given diameter. Problemy Peredachi Informatsii, pages 106-109, 1987. On ResearchGate.
G. Katona, Intersection theorems for systems of finite sets, Acta Mathematica Academiae Scientiarum Hungaricae 15, 329-337 (1964).
Daniel J. Kleitman, On a combinatorial conjecture of Erdös, Journal of Combinatorial Theory, Series A 1, 209-214, (1966).
FORMULA
EXAMPLE
The array begins:
1, 1, 2, 1, 0, 1, 2, 1, ...
0, 1, 2, 2, 2, 2, 2, 2, ...
0, 1, 2, 3, 4, 4, 4, 4, ...
0, 1, 2, 4, 6, 7, 8, 8, ...
0, 1, 2, 5, 8, 11, 14, 15, ...
0, 1, 2, 6, 10, 16, 22, 26, ...
0, 1, 2, 7, 12, 22, 32, 42, ...
0, 1, 2, 8, 14, 29, 44, 64, ...
...
MATHEMATICA
A[n_, k_]:=If[OddQ[k], Sum[Binomial[n, i], {i, 0, (k-1)/2}], Binomial[n-1, k/2-1]+Sum[Binomial[n, i], {i, 0, k/2-1}]]; Table[A[n-k, k], {n, 0, 12}, {k, 0, n}]//Flatten
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Stefano Spezia, May 19 2024
STATUS
approved