A001116 - OEIS

%I M1585 N0617 #69 Nov 08 2024 08:37:25

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

%N Maximal kissing number of an n-dimensional lattice.

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

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

%C From _Nathan L. Skirrow_, Jun 04 2023: (Start)

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

%C 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)

%D J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, 3rd. ed., 1993. p. 15.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D 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.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Henry Cohn and Anqi Li, <a href="https://arxiv.org/abs/2411.04916">Improved kissing numbers in seventeen through twenty-one dimensions</a>, arXiv:2411.04916 [math.MG], 2024.

%H J. H. Conway and N. J. A. Sloane, <a href="http://dx.doi.org/10.1007/978-1-4757-2016-7">Sphere Packings, Lattices and Groups</a>, Springer-Verlag, p. 31-62.

%H J. Leech and N. J. A. Sloane, <a href="https://doi.org/10.1090/S0002-9904-1970-12533-2">New sphere packings in dimensions 9-15</a>, Bull. Amer. Math. Soc., 76 (1970), 1006-1010.

%H G. Nebe and N. J. A. Sloane, <a href="http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/kiss.html">Table of highest kissing numbers known</a>

%H N. J. A. Sloane, <a href="http://neilsloane.com/doc/sg.txt">My favorite integer sequences</a>, in Sequences and their Applications (Proceedings of SETA '98).

%H N. J. A. Sloane, <a href="http://neilsloane.com/doc/g4g7.pdf">Seven Staggering Sequences</a>.

%H N. J. A. Sloane, <a href="https://arxiv.org/abs/2301.03149">"A Handbook of Integer Sequences" Fifty Years Later</a>, arXiv:2301.03149 [math.NT], 2023, p. 21.

%H John Tangen, <a href="/A259343/a259343.pdf">Letter to N. J. A. Sloane, Apr 27 1978</a>

%H G. L. Watson, <a href="https://eudml.org/doc/268650">The number of minimum points of a positive quadratic form</a>, Dissertationes Math., 84 (1971), 42 pp.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/KissingNumber.html">Kissing Number.</a>

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

%e 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.

%Y Cf. A002336, A028923, A257479.

%K nonn,nice,hard,more

%O 0,2

%A _N. J. A. Sloane_