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%I #31 Mar 13 2018 22:13:49
%S 0,1,3,6,9,12,15,18,21
%N Maximal contact number of a subset of n balls from the face-centered cubic lattice.
%C 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.
%C a(13) >= 36 (take one ball and its 12 neighbors), so this is different from A008486.
%C 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]
%C 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
%H Bezdek, Karoly, <a href="http://dx.doi.org/10.1007/s00454-012-9405-9">Contact Numbers for Congruent Sphere Packings in Euclidean 3-Space</a>, Discrete Comput. Geom. 48 (2012), no. 2, 298--309. MR2946449
%H K. Bezdek, M. A. Khan, <a href="https://arxiv.org/abs/1601.00145">Contact number for sphere packings</a>, arXiv:1601.00145 [math.MG], 2016.
%H K. Bezdek, S. Reid, <a href="http://dx.doi.org/10.1007/s00022-013-0156-4">Contact graphs of unit sphere packings revisited</a>, J. Geom. 104 (1) (2013) 57-83.
%H J. P. K. Doye, D. J. Wales, <a href="https://doi.org/10.1016/S0009-2614(95)01223-0">Magic numbers and growth sequences of small face-centered-cubic and decahedral clusters</a>, Chem. Phys. Lett. 247 (1995) 339, Table 1 column n(fcc).
%H G. Nebe and N. J. A. Sloane, <a href="http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/D3.html">Home page for this lattice</a>
%H <a href="/index/Fa#fcc">Index entries for sequences related to f.c.c. lattice</a>
%Y Cf. A004015, A005901, A038173.
%K nonn,more
%O 1,3
%A _N. J. A. Sloane_, Jul 31 2012