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A257479 Maximal kissing number in n dimensions: maximal number of unit spheres that can touch another unit sphere. 3
2, 6, 12, 24 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Two additional dimensions have been solved: a(8) = 240, a(24) = 196560 [Odlyzko and Sloane; Levenstein].

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

Table of n, a(n) for n=1..4.

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.

Mittelmann, Hans D.; Vallentin, Frank, "High accuracy semidefinite programming bounds for kissing numbers." 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.

Wikipedia, Kissing number problem

EXAMPLE

For a(2), the maximal number of pennies that can tough 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: A192703 A192969 A291014 * A244043 A058868 A129314

Adjacent sequences:  A257476 A257477 A257478 * A257480 A257481 A257482

KEYWORD

nonn,bref

AUTHOR

Peter Woodward, Apr 25 2015

EXTENSIONS

Entry revised by N. J. A. Sloane, May 08 2015

STATUS

approved

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Last modified September 26 10:58 EDT 2017. Contains 292518 sequences.