A001116 Maximal kissing number of an n-dimensional lattice. (Formerly M1585 N0617)

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

Offset

0,2

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 Nathan 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)

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

Table of n, a(n) for n=0..9.

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.

Formula

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

Example

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.

Crossrefs

Cf. A002336, A028923, A257479.

Sequence in context: A211978 A028923 A187272 * A002336 A030625 A029929

Adjacent sequences: A001113 A001114 A001115 * A001117 A001118 A001119

Keyword

nonn,nice,hard,more,changed

Author

N. J. A. Sloane

Status

approved