A001116 -id:A001116 - cp's OEIS frontend

A259343 A001116(n) + 1.

Original entry on oeis.org

1, 3, 7, 13, 25, 41, 73, 127, 241, 273

Offset: 0

N. J. A. Sloane, Jun 27 2015, following a suggestion from John Tangen, Apr 27 1978

John Tangen, Letter to N. J. A. Sloane, Apr 27 1978

Cf. A001116.

A002336 Maximal kissing number of n-dimensional laminated lattice.

Original entry on oeis.org

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.

A257479 Maximal kissing number in n dimensions: maximal number of unit spheres that can touch another unit sphere.

Original entry on oeis.org

2, 6, 12, 24

Offset: 1

Peter Woodward, Apr 25 2015

Two additional terms are known: a(8) = 240, a(24) = 196560 [Odlyzko and Sloane; Levenshtein].
Lower bounds for a(5) onwards are 40, 72, 126, 240 (exact), 306, 500, ... - N. J. A. Sloane, May 15 2015
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
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

			For a(2), the maximal number of pennies that can touch one penny is 6.
For a(3), the most spheres that can simultaneously touch a central sphere of the same radius is 12.

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, 3rd. ed., Chap. 3, esp. pp. xxi, 23, etc.
Musin, Oleg Rustumovich. "The problem of the twenty-five spheres." Russian Mathematical Surveys 58.4 (2003): 794-795.

Eiichi Bannai and N. J. A. Sloane, Uniqueness of certain spherical codes, Canad. J. Math. 33 (1981), no. 2, 437-449.
J. Leech, The problem of the thirteen spheres, Math. Gaz., 40 (1956), 22-23.
V. I. Levenshtein, On bounds for packings in n-dimensional Euclidean space, Dokl. Akad. Nauk., 245 (1979), 1299-1303; English translation in Soviet Math. Doklady, 20 (1979), 417-421.
Hans D. Mittelmann and Frank Vallentin, High accuracy semidefinite programming bounds for kissing numbers, arXiv:0902.1105 [math.OC], 2009; Exp. Math. (2009), no. 19, 174-178.
G. Nebe and N. J. A. Sloane, Table of highest kissing numbers known
A. M. Odlyzko and N. J. A. Sloane, New bounds on the number of unit spheres that can touch a unit sphere in n dimensions, J. Combin. Theory Ser. A 26 (1979), no. 2, 210-214.
N. J. A. Sloane, "A Handbook of Integer Sequences" Fifty Years Later, arXiv:2301.03149 [math.NT], 2023, p. 21.
Wikipedia, Kissing number problem

Cf. A001116 (n-dimensional lattice), A002336 (n-dimensional laminated lattice), A028923 (n-dimensional lattice Kappa_n).

Cf. A008408.

Entry revised by N. J. A. Sloane, May 08 2015

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

A120701 Number of unit circles which fit touching a circle of radius n-1, i.e., with their centers on a circle of radius n.

Original entry on oeis.org

2, 6, 9, 12, 15, 18, 21, 25, 28, 31, 34, 37, 40, 43, 47, 50, 53, 56, 59, 62, 65, 69, 72, 75, 78, 81, 84, 87, 91, 94, 97, 100, 103, 106, 109, 113, 116, 119, 122, 125, 128, 131, 135, 138, 141, 144, 147, 150, 153, 157, 160, 163, 166, 169, 172, 175, 179, 182, 185, 188

Offset: 1

Martin Fuller, Jun 28 2006

Coincides with A022844 = floor(n*Pi) except at n=1, 25510582, ... (sequence A120702).

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A001116, A002486, A022844, A120702.

R:= RealField(30); [Floor(Pi(R)/Arcsin(1/n)) : n in [1..70]]; // G. C. Greubel, Aug 25 2023

Table[Floor[Pi/ArcSin[1/n]], {n, 60}] (* Indranil Ghosh, Jul 21 2017 *)

from mpmath import mp, pi, asin
mp.dps=100
def a(n): return int(floor(pi/asin(1./n)))
print([a(n) for n in range(1, 61)]) # Indranil Ghosh, Jul 21 2017

[floor(pi/arcsin(1/n)) for n in range(1,71)] # G. C. Greubel, Aug 25 2023

a(n) = floor(Pi/arcsin(1/n)).

A028925 Maximal number of pairs of minimal vectors in an n-dimensional lattice.

Original entry on oeis.org

0, 1, 3, 6, 12, 20, 36, 63, 120, 136

Offset: 0

N. J. A. Sloane

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 15.

A001116 / 2.

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

cp's OEIS Frontend

A259343 A001116(n) + 1.

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A002336 Maximal kissing number of n-dimensional laminated lattice.

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A257479 Maximal kissing number in n dimensions: maximal number of unit spheres that can touch another unit sphere.

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A028923 Kissing number of n-dimensional lattice Kappa_n.

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A120701 Number of unit circles which fit touching a circle of radius n-1, i.e., with their centers on a circle of radius n.

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A028925 Maximal number of pairs of minimal vectors in an n-dimensional lattice.

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

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