cp's OEIS Frontend

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

Terms for n > 4 have not been determined so far. Alfredo Andreini in 1905 gave a value of 25 for a(3), later found to be incorrect. The value 28 for a(3) was given by Norman Johnson in 1991 and later in 1994 independently by Branko Grünbaum. The value for a(4) was given by George Olshevsky in 2006.

Deza and Shtogrin (2000) agree that the value of a(3) is 28, although the authors do not provide a proof. - Felix Fröhlich, Nov 29 2014

From Felix Fröhlich, Feb 03 2019: (Start)

The 11 convex uniform tilings are all illustrated in Kepler, 1619. For an argument that exactly 11 such tilings exist, see Grünbaum, Shephard, 1977.

In dimension 2, the definition of "uniform polytope" usually seems to be equivalent to the regular polygons in order to exclude polygons that alternate two different edge-lengths. Applying this principle retroactively to dimension 1 (as done, as I assume, by Coxeter, see Coxeter, 1973, p. 129) yields a(1) = 1. (End)

Hurdle races not included.

The marathon distance was only standardized to 42195 meters in 1924; before then, distances varied from about 40000 to 40750. This sequence does not include hurdles, steeplechase, relays, medleys, team races, walks, or cross country. It also excludes the 60 meter of the 1900 and 1904 Olympics and the women's 3000 meter in 1984, 1988, and 1992; additionally, the women's running events did not include all of these distances until 1996. - Charles R Greathouse IV, Jan 06 2016