This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A257479 #70 Jan 12 2025 04:51:24
%S A257479 2,6,12,24
%N A257479 Maximal kissing number in n dimensions: maximal number of unit spheres that can touch another unit sphere.
%C A257479 Two additional terms are known: a(8) = 240, a(24) = 196560 [Odlyzko and Sloane; Levenshtein].
%C A257479 Lower bounds for a(5) onwards are 40, 72, 126, 240 (exact), 306, 500, ... - _N. J. A. Sloane_, May 15 2015
%C A257479 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 an 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
%C A257479 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 an 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
%D A257479 J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, 3rd. ed., Chap. 3, esp. pp. xxi, 23, etc.
%D A257479 Musin, Oleg Rustumovich. "The problem of the twenty-five spheres." Russian Mathematical Surveys 58.4 (2003): 794-795.
%H A257479 Eiichi Bannai and N. J. A. Sloane, <a href="http://dx.doi.org/10.4153/CJM-1981-038-7">Uniqueness of certain spherical codes</a>, Canad. J. Math. 33 (1981), no. 2, 437-449.
%H A257479 J. Leech, <a href="http://www.jstor.org/stable/3610264">The problem of the thirteen spheres</a>, Math. Gaz., 40 (1956), 22-23.
%H A257479 V. I. Levenshtein, <a href="https://doi.org/10.4213/rm651">On bounds for packings in n-dimensional Euclidean space</a>, Dokl. Akad. Nauk., 245 (1979), 1299-1303; English translation in Soviet Math. Doklady, 20 (1979), 417-421.
%H A257479 Hans D. Mittelmann and Frank Vallentin, <a href="https://arxiv.org/abs/0902.1105">High accuracy semidefinite programming bounds for kissing numbers</a>, arXiv:0902.1105 [math.OC], 2009; Exp. Math. (2009), no. 19, 174-178.
%H A257479 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 A257479 A. M. Odlyzko and N. J. A. Sloane, <a href="https://doi.org/10.1016/0097-3165(79)90074-8">New bounds on the number of unit spheres that can touch a unit sphere in n dimensions</a>, J. Combin. Theory Ser. A 26 (1979), no. 2, 210-214.
%H A257479 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 A257479 Wikipedia, <a href="http://en.wikipedia.org/wiki/Kissing_number_problem">Kissing number problem</a>
%e A257479 For a(2), the maximal number of pennies that can touch one penny is 6.
%e A257479 For a(3), the most spheres that can simultaneously touch a central sphere of the same radius is 12.
%Y A257479 Cf. A001116 (n-dimensional lattice), A002336 (n-dimensional laminated lattice), A028923 (n-dimensional lattice Kappa_n).
%Y A257479 Cf. A008408.
%K A257479 nonn,bref
%O A257479 1,1
%A A257479 _Peter Woodward_, Apr 25 2015
%E A257479 Entry revised by _N. J. A. Sloane_, May 08 2015