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A214813
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Maximal contact number of a subset of n balls from the face-centered cubic lattice.
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1
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OFFSET
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1,3
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COMMENTS
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If S is an arrangement of non-overlapping balls of radius 1, the contact number of S is the number of pairs of balls that just touch each other.
a(13) >= 36 (take one ball and its 12 neighbors), so this is different from A008486.
If b(n) denotes the maximal contact number of any arrangement of n balls then it is conjectured that a(n) = b(n) for n <= 9. It is also known that b(10)>=25, b(11)>=29, b(12)>=33 and of course b(13) >= a(13) >= 36. [Bezdek 2012]
Note that Figure 1e of Bezdek's arxiv:1601.00145 shows at n=5 a sphere packing with 9 contacts on the hexagonal close package (!), not on the cubic close package (which equals the f.c.c.). [In Figure 1e there is one sphere that touches from above a set of 3 spheres in a middle layer right above the bottom sphere; so this needs the ABABA... layer structures of the h.c.p, and cannot be done with the ABCABC... layer structure of the f.c.c.] So Figure 1e is not demonstrating a(5)=9. The correct value for the f.c.c is apparently a(5)=8 (where two structures with 8 contacts exist.) - R. J. Mathar, Mar 13 2018
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LINKS
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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STATUS
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approved
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