A386400 Minimum volume of n polycubes such that each polycube is adjacent to every other polycube.
1, 2, 4, 7, 13, 20, 28, 36
Offset: 1
Examples
a(n) can be seen by the following diagrams, where each cell in the i-th polycube is labeled by [i]. a(1)=1: [1] a(2)=2: [1][2] a(3)=4: [1][3] [2][3] a(4)=7: Layer 1 Layer 2 [1][3] [4][4] [2][3] [2] a(5)=13: Layer 1 Layer 2 [5][1][3] [4][4][4] [5][2][3] [2] [5][3][3]
Links
- Wikipedia, Hadwiger number.
Crossrefs
Cf. A193416.
Formula
a(n) <= n^2-n-6, for n>4
This upper bound comes from the following construction which holds for all n>4:
Layer 1 Layer 2
[1][1]... [1] [1] [1] [1] [n]
[1][4]...[n-3][n-2][n-1] [2][2]... [2] [2] [2] [n]
[3][4]...[n-3][n-2][n-1] [3][3]... [3] [3] [3] [n]
[4]...[n-3][n-2][n-1] [4]... [4] [4] [4] [n]
... : : : ... : : : :
[n-3][n-2][n-1] [n-3][n-3][n-3][n]
[n-2][n-1] [n-2] [n] [n]
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