A257479 -id:A257479 - cp's OEIS frontend

A002336 Maximal kissing number of n-dimensional laminated lattice.

0, 2, 6, 12, 24, 40, 72, 126, 240, 272, 336, 438, 648, 906, 1422, 2340, 4320, 5346, 7398, 10668, 17400, 27720, 49896, 93150, 196560, 196656

Offset: 0

N. J. A. Sloane and J. H. Conway

This sequence is concerned with lattice packings. For unrestricted packings the values are presently known only in dimensions 1, 2, 3, 4, 8 and 24: 2, 6, 12, 24, 240, 196560 (cf. A257479). See Conway and Sloane for details.

J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Springer-Verlag, p. 174.
C. Musès, The dimensional family approach in (hyper)sphere packing..., Applied Math. Computation 88 (1997), pp. 1-26.
G. Nebe and N. J. A. Sloane, Table of highest kissing numbers known

Cf. A001116, A028923, A257479.

a(n) <= A001116(n).

In dimensions 25-32 the highest kissing numbers presently known for laminated lattices are 196848, 197142, 197736, 198506, 200046, 202692, 208320.

A001116 Maximal kissing number of an n-dimensional lattice.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

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)

			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.

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).

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.

Cf. A002336, A028923, A257479.

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

A217695 Decimal expansion of largest angular separation (in radians) between 13 points on a unit sphere.

Original entry on oeis.org

9, 9, 7, 2, 2, 3, 5, 9, 2, 4, 3, 8, 1, 1, 9, 1, 6, 3, 6, 5, 4, 7, 7, 0, 4, 5, 0, 5, 7, 6, 1, 2, 2, 0, 1, 4, 5, 5, 0, 3, 2, 4, 4, 9, 3, 7, 3, 3, 0, 1, 4, 4, 2, 5, 3, 4, 6, 2, 8, 1, 0, 3, 4, 1, 6, 8, 4, 0, 0, 7, 3, 5, 2, 1, 1, 1, 8, 0, 5, 4, 5, 4, 4, 3, 0, 0, 7, 8, 5, 6, 8, 8, 1, 2, 1, 2, 6, 0, 2, 2, 8

Offset: 0

Charles R Greathouse IV, Mar 20 2013

Since this is less than Pi/3, the kissing number in three dimensions is 12 rather than 13. Related to the Tammes problem.

			0.99722359243811916365477045057612201455032449373301442534628103416840073521118... radians = 57.1367030... degrees.

Oleg Musin and Alexey Tarasov, The strong thirteen spheres problem, Discrete & Computational Geometry 48:1 (2012), pp. 128-141. doi:10.1007/s00454-011-9392-2

Cf. A257479.

digits = 101; x0 = x /. FindRoot[2*Tan[3*Pi/8-x/4]-(1-2*Cos[x])/Cos[x]^2 == 0, {x, 6/5}, WorkingPrecision -> digits+1]; ArcCos[Cos[x0]/(1-Cos[x0])] // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Feb 20 2014, after PARI *)

(a->acos(cos(a)/(1-cos(a))))(solve(x=1,2,2*tan(3*Pi/8-x/4)-(1-2*cos(x))/cos(x)^2))

A028923 Kissing number of n-dimensional lattice Kappa_n.

Original entry on oeis.org

0, 2, 6, 12, 24, 40, 72, 92, 132, 180, 276, 432, 756, 918, 1232, 1746, 3290, 4266

Offset: 0

N. J. A. Sloane

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 161.

G. Nebe and N. J. A. Sloane, Table of highest kissing numbers known

Cf. A001116, A002336, A257479.

a(n) >= A001116(n). - Charles R Greathouse IV, Mar 25 2022

A127081 One-sided kissing number for spheres in n-dimensional Euclidean space.

Original entry on oeis.org

1, 4, 9, 18

Offset: 1

Jonathan Vos Post, Mar 21 2007

a(8) = 183. Musin's conjectures: a(5) = 32, a(24) = 144855.

"Let H be a closed half-space of n-dimensional Euclidean space. Suppose S is a unit sphere in H that touches the supporting hyperplane of H. The one-sided kissing number B(n) is the maximal number of unit nonoverlapping spheres in H that can touch S." [Musin]

Oleg R. Musin, The one-sided kissing number in four dimensions, arXiv:math/0511071 [math.MG], 2007.
Oleg R. Musin, Five Essays on the Geometry of László Fejes Tóth, arXiv:1604.02776 [math.MG], 2016-2017.

Cf. A001116, A257479.

Edited by N. J. A. Sloane, Mar 23 2007

Conjectured a(5) removed from Data by Andrey Zabolotskiy, Sep 22 2021

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A002336 Maximal kissing number of n-dimensional laminated lattice.

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A001116 Maximal kissing number of an n-dimensional lattice.

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A217695 Decimal expansion of largest angular separation (in radians) between 13 points on a unit sphere.

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A028923 Kissing number of n-dimensional lattice Kappa_n.

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A127081 One-sided kissing number for spheres in n-dimensional Euclidean space.

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