Seven Staggering Sequences

When my "Handbook of Integer Sequences" came out in 1973, Philip Morrison gave it an enthusiastic review in the Scientific American and Martin Gardner was kind enough to say in his Mathematical Games column that "every recreational mathematician should buy a copy forthwith." That book contained 2372 sequences. Today the "On-Line Encyclopedia of Integer Sequences" contains 117000 sequences. This paper will describe seven that I find especially interesting. These are the EKG sequence, Gijswijt's sequence, a numerical analog of Aronson's sequence, approximate squaring, the integrality of n-th roots of generating functions, dissections, and the kissing number problem. (Paper for conference in honor of Martin Gardner's 91st birthday.)Comment: 12 pages. A somewhat different version appeared in "Homage to a Pied Puzzler", E. Pegg Jr., A. H. Schoen and T. Rodgers (editors), A. K. Peters, Wellesley, MA, 2009, pp. 93-11

Rigidity of spherical codes

A packing of spherical caps on the surface of a sphere (that is, a spherical code) is called rigid or jammed if it is isolated within the space of packings. In other words, aside from applying a global isometry, the packing cannot be deformed. In this paper, we systematically study the rigidity of spherical codes, particularly kissing configurations. One surprise is that the kissing configuration of the Coxeter-Todd lattice is not jammed, despite being locally jammed (each individual cap is held in place if its neighbors are fixed); in this respect, the Coxeter-Todd lattice is analogous to the face-centered cubic lattice in three dimensions. By contrast, we find that many other packings have jammed kissing configurations, including the Barnes-Wall lattice and all of the best kissing configurations known in four through twelve dimensions. Jamming seems to become much less common for large kissing configurations in higher dimensions, and in particular it fails for the best kissing configurations known in 25 through 31 dimensions. Motivated by this phenomenon, we find new kissing configurations in these dimensions, which improve on the records set in 1982 by the laminated lattices.Comment: 39 pages, 8 figure

Sphere packings revisited

AbstractIn this paper we survey most of the recent and often surprising results on packings of congruent spheres in d-dimensional spaces of constant curvature. The topics discussed are as follows:–Hadwiger numbers of convex bodies and kissing numbers of spheres;–touching numbers of convex bodies;–Newton numbers of convex bodies;–one-sided Hadwiger and kissing numbers;–contact graphs of finite packings and the combinatorial Kepler problem;–isoperimetric problems for Voronoi cells, the strong dodecahedral conjecture and the truncated octahedral conjecture;–the strong Kepler conjecture;–bounds on the density of sphere packings in higher dimensions;–solidity and uniform stability.Each topic is discussed in details along with some of the “most wanted” research problems

N.J.A.: On Kissing Numbers in Dimensions 32 to 128

An elementary construction using binary codes gives new record kissing numbers in dimensions Let τn denote the maximal kissing number in dimension n, that is, the greatest number of n-dimensional spheres that can touch another sphere of the same size. Although asymptotic bounds on τn are known [5], little is known about explicit constructions, especially for n> 32. Up to now the best explicit constructions have come from lattice packings. The kissing numbe