

A214813


Maximal contact number of a subset of n balls from the facecentered cubic lattice.


1




OFFSET

1,3


COMMENTS

If S is an arrangement of nonoverlapping 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


LINKS

Table of n, a(n) for n=1..9.
Bezdek, Karoly, Contact Numbers for Congruent Sphere Packings in Euclidean 3Space, Discrete Comput. Geom. 48 (2012), no. 2, 298309. MR2946449
K. Bezdek, M. A. Khan, Contact number for sphere packings, arXiv:1601.00145 [math.MG], 2016.
K. Bezdek, S. Reid, Contact graphs of unit sphere packings revisited, J. Geom. 104 (1) (2013) 5783.
J. P. K. Doye, D. J. Wales, Magic numbers and growth sequences of small facecenteredcubic and decahedral clusters, Chem. Phys. Lett. 247 (1995) 339, Table 1 column n(fcc).
G. Nebe and N. J. A. Sloane, Home page for this lattice
Index entries for sequences related to f.c.c. lattice


CROSSREFS

Cf. A004015, A005901, A038173.
Sequence in context: A198263 A276192 A282143 * A119888 A305495 A003252
Adjacent sequences: A214810 A214811 A214812 * A214814 A214815 A214816


KEYWORD

nonn,more


AUTHOR

N. J. A. Sloane, Jul 31 2012


STATUS

approved



