login
A257479
Maximal kissing number in n dimensions: maximal number of unit spheres that can touch another unit sphere.
5
OFFSET
1,1
COMMENTS
Two additional terms are known: a(8) = 240, a(24) = 196560 [Odlyzko and Sloane; Levenshtein].
Lower bounds for a(5) onwards are 40, 72, 126, 240 (exact), 306, 500, ... - N. J. A. Sloane, May 15 2015
It seems that, while n is even, a lower bound for a(n) can be written as f(n) = (2n + 2^n) - k(n)^2, where k(2) = 2, for n > 2, k(n) = 2^(n/2) - q, q = {2^t, 3*2^t}, t is integer, t > 0, 2 <= q <= n, and f(n) <= a(n) (Note: for n <= 24, q = n at n = {4, 6, 8, 24}, q = 0 at n = 2). - Sergey Pavlov, Mar 17 2017
It also seems that, while n is even, an upper bound for a(n) can be written as f(n) = (2n + 2^n) - k(n)^2, where k(2) = 0, for n > 2, k(n) = 2^(n/2) - q, q = {2^t, 3*2^t}, t is integer, t > 0, n <= q <= 2n, and f(n) >= a(n) (Note: for n <= 24, q = n at n = {2, 4, 12, 16}). - Sergey Pavlov, Mar 19 2017
REFERENCES
J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, 3rd. ed., Chap. 3, esp. pp. xxi, 23, etc.
Musin, Oleg Rustumovich. "The problem of the twenty-five spheres." Russian Mathematical Surveys 58.4 (2003): 794-795.
LINKS
Eiichi Bannai and N. J. A. Sloane, Uniqueness of certain spherical codes, Canad. J. Math. 33 (1981), no. 2, 437-449.
J. Leech, The problem of the thirteen spheres, Math. Gaz., 40 (1956), 22-23.
V. I. Levenshtein, On bounds for packings in n-dimensional Euclidean space, Dokl. Akad. Nauk., 245 (1979), 1299-1303; English translation in Soviet Math. Doklady, 20 (1979), 417-421.
Hans D. Mittelmann and Frank Vallentin, High accuracy semidefinite programming bounds for kissing numbers, arXiv:0902.1105 [math.OC], 2009; Exp. Math. (2009), no. 19, 174-178.
G. Nebe and N. J. A. Sloane, Table of highest kissing numbers known
A. M. Odlyzko and N. J. A. Sloane, New bounds on the number of unit spheres that can touch a unit sphere in n dimensions, J. Combin. Theory Ser. A 26 (1979), no. 2, 210-214.
N. J. A. Sloane, "A Handbook of Integer Sequences" Fifty Years Later, arXiv:2301.03149 [math.NT], 2023, p. 21.
EXAMPLE
For a(2), the maximal number of pennies that can touch one penny is 6.
For a(3), the most spheres that can simultaneously touch a central sphere of the same radius is 12.
CROSSREFS
Cf. A001116 (n-dimensional lattice), A002336 (n-dimensional laminated lattice), A028923 (n-dimensional lattice Kappa_n).
Cf. A008408.
Sequence in context: A294532 A323950 A291014 * A307740 A244043 A058868
KEYWORD
nonn,bref
AUTHOR
Peter Woodward, Apr 25 2015
EXTENSIONS
Entry revised by N. J. A. Sloane, May 08 2015
STATUS
approved