A001116 Maximal kissing number of an n-dimensional lattice.

0, 2, 6, 12, 24, 40, 72, 126, 240, 272

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• hard
• more

Comments

a(9) = 272 was determined by Watson (1971). a(10) is probably 336.

Lower bounds for the next 4 terms are 336, 438, 756, 918.

From Natalia L. Skirrow, Jun 04 2023: (Start)

Trivial upper bounds given by A257479 are 553, 869, 1356, 2066.

a(n) coincides with A257479(n) when a lattice achieves the non-lattice-constrained kissing number, for a(0)=0, a(1)=2, a(2)=6 (A_2), a(3)=12 (A_3), a(4)=24 (D_4), a(8)=240 (E_8) and a(24)=196560 (Leech). A002336(n) agrees with a(n) for all n<=9 (and equality is unknown thereafter), and A028923(n)=a(n) iff n<=6. (End)

Examples

			In three dimensions, each sphere in the face-centered cubic lattice D_3 touches 12 others, and the kissing number in any other three-dimensional lattice is less than 12.
		

References

• J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, 3rd. ed., 1993. p. 15.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane, Seven Staggering Sequences, in Homage to a Pied Puzzler, E. Pegg Jr., A. H. Schoen and T. Rodgers (editors), A. K. Peters, Wellesley, MA, 2009, pp. 93-110.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Henry Cohn and Anqi Li, Improved kissing numbers in seventeen through twenty-one dimensions, arXiv:2411.04916 [math.MG], 2024.
• J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Springer-Verlag, p. 31-62.
• J. Leech and N. J. A. Sloane, New sphere packings in dimensions 9-15, Bull. Amer. Math. Soc., 76 (1970), 1006-1010.
• G. Nebe and N. J. A. Sloane, Table of highest kissing numbers known
• N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).
• N. J. A. Sloane, Seven Staggering Sequences.
• N. J. A. Sloane, "A Handbook of Integer Sequences" Fifty Years Later, arXiv:2301.03149 [math.NT], 2023, p. 21.
• John Tangen, Letter to N. J. A. Sloane, Apr 27 1978
• G. L. Watson, The number of minimum points of a positive quadratic form, Dissertationes Math., 84 (1971), 42 pp.
• Eric Weisstein's World of Mathematics, Kissing Number.

Crossrefs

Cf. A002336, A028923, A257479.

Formula

A002336(n),A028923(n) <= a(n) <= A257479(n).