A084616 -id:A084616 - cp's OEIS frontend

A084617 Maximum number of circles with diameter 1 that can be packed in a square of side length n.

Original entry on oeis.org

1, 4, 9, 16, 25, 36, 49, 68, 86, 106, 128, 152, 181, 216, 247, 280

Offset: 1

Hugo Pfoertner, Jun 01 2003

For n>5 the sequence terms are only conjectures. For more information see comment given in A084616.

See under A084616.

M. Goldberg, The packing of equal circles in a square, Math. Mag. 43 (1) (1970), 24-30
Packomania, Packing Circles in a Square.

Cf. A084616, A051657, A023393.

More terms from Sergio Pimentel, Aug 08 2006

A084618 Maximum number of circles of area 1 that can be packed in a circle of area n.

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 4, 5, 7, 7, 8, 8, 8, 9, 10, 11, 12, 13, 14, 14, 15, 16, 17, 19, 19, 19, 20, 21, 21, 22, 23, 24, 24, 26, 27, 27, 28, 29, 30, 31, 31, 32, 33, 34, 35, 37, 37, 37, 38, 39, 40, 40, 41, 42, 42, 43, 44, 45, 46, 47, 48, 48, 49, 50, 51, 52, 53, 55, 55, 55, 56, 57, 58, 59

Offset: 1

Hugo Pfoertner, Jun 01 2003

For most values of n these are only conjectures, supported by numerical results.

			a(4)=2 because a circle of area 4 is the smallest one covering two circles of area 1.
a(9)=7 is the arrangement of 6 circles closely packed around another circle. This arrangement fits into a circle that has 3*radius of smaller circles and thus 9*their area.

For list of references given by E. Specht, see corresponding link.

Erich Friedman, Circles in Circles
Hugo Pfoertner, Minimum area of circle needed to cover n circles of area 1
E. Specht, The best known packings of equal circles in the unit circle

Cf. A084616, A023393, A084644.

Equivalent sequences for packing into a square: A337020, and equilateral triangle: A337019.

A084824 Maximum number of spheres of diameter one that can be packed in a cube of volume n (i.e., with edge length n^(1/3)).

Original entry on oeis.org

1, 1, 1, 2, 4, 4, 5, 8, 8, 8, 9, 9, 10, 11, 14, 14, 14, 15, 18, 18, 19, 19, 21, 21, 23, 24, 27, 27, 27, 27, 32, 32, 32, 33

Offset: 1

Hugo Pfoertner, Jun 12 2003

Higher sequence terms are only conjectures found by numerical experimentation.

			a(5) = 4 because a cube of edge length 5^(1/3) = 1.7099759 is large enough to contain 4 spheres arranged as a tetrahedron, which requires a minimum enclosing cube of edge length 1+sqrt(2)/2 = 1.70710678.

Dave Boll, Optimal Packing of Circles and Spheres
Thierry Gensane, Dense Packings of Equal Spheres in a Cube, The Electronic Journal of Combinatorics 11 (2004), #R33.
M. Goldberg, On the Densest Packing of Equal Spheres in a Cube, Math. Mag. 44, 199-208, 1971.
Hugo Pfoertner, Best packing of equal spheres in a cube. Numerical results.
Hugo Pfoertner, Densest Packings of Equal Spheres in a Cube. Visualizations.
J. Schaer, On the Densest Packing of Spheres in a Cube, Can. Math. Bul. 9, 265-270, 1966.

Cf. A084825, A084826, A084827, A084616.

Corrected erroneous a(14) and extended to a(34) by Hugo Pfoertner, including results from Thierry Gensane, Jun 23 2011

A253570 Maximum number of circles of radius 1 that can be packed into a regular n-gon with side length 2 (conjectured).

Original entry on oeis.org

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

Offset: 3

Felix Fröhlich, Jan 03 2015

The values were obtained by constructing the circle arrangements in a vector graphics program and have not been proved to be correct.
From David Consiglio, Jr., Jan 09 2015: (Start)
As n increases, the n-gon more and more closely approximates a circle. As a result, the lower bound (which is highly likely to be the correct term for larger and larger n) is the number of circles that can be packed into an inscribed circle, the radius of which is given by the expression cot(Pi/n). Look up this radius in column 3 at www.packomania.com to find the lower bound of a(n).
A rough upper bound would be the closest packing of circles into the area of the n-gon (formula below). A better upper bound is likely possible.
See file for lower and upper bounds through a(20).  The lower bounds have been proved for a(3) through a(13).
(End)

David Consiglio, Jr., Lower and Upper Bounds
Felix Fröhlich, Illustration of circle arrangements associated with a(3)-a(12)
E. Specht, The best known packings of equal circles in a circle (complete up to N = 2600), Packomania.

Cf. A023393, A051657, A084616.

Upper bound = floor(n/(2*sqrt(3)*tan(Pi/n))).

A337020 Maximum number of circles with unit area that can be packed into a square with an area of n.

Original entry on oeis.org

0, 1, 1, 2, 3, 4, 4, 5, 5, 6, 8, 9, 9, 9, 10, 11, 12, 13, 13, 15, 16, 16, 16, 18, 18, 20, 20, 21, 22, 22, 24, 25, 25, 25, 27, 27, 28, 30, 30

Offset: 1

Ya-Ping Lu, Nov 06 2020

Terms beyond a(n) = 30 (n = 38 & 39) except a(n) = 36 are conjectures supported by numerical results (see Packomania in the links) and terms for n from 40 through 70 are: 31, 31, 33, 33, 34, 35, 36, 36, 38, 39, 39, 40, 41, 42, 42, 43, 44, 44, 46, 46, 47, 48, 49, 50, 52, 52, 53, 53, 54, 55, 56.
The packing density, a(n)/n, approaches sqrt(3)*Pi/6 as n tends to infinity.
References for the known optimal packings are given in Table 1.2 on page 10 and the bibliography on pages 219-225 of the book by Szabo et al. (see References).

P. G. Szabo, M. Cs. Markot, T. Csendes, E. Specht, L. G. Casado, and I. Garcia, New Approaches to Circle Packing in a Square, Springer, 2007.

R. L. Graham and B. D. Lubachevsky, Repeated Patterns of Dense Packings of Equal Disks in a Square, Electronic Journal of Combinatorics 3 (1996), #R16; arXiv:math/0406394 [math.MG], 2004.
Eckard Specht, editor, Packomania, Section 1: Packings of equal and unequal circles in fixed-sized containers with maximum packing density.

Cf. A084616, A084617, A084644, A093766, A337019.

A308578 Maximum number of non-overlapping circles of radius 1/n that can be placed inside a unit square.

Original entry on oeis.org

0, 1, 1, 4, 5, 9, 10, 16, 20, 25, 30, 36

Offset: 1

Ethan D. Kidd, Jun 08 2019

Alternatively described as the maximum number of circles of unit radius that can be placed inside a square of side length n.

It appears that the terms a(2) to a(8) are equal to the related terms of A189889.

			a(6)=9 because 9 circles of radius 1/6 can be placed in a 3 X 3 regular grid inside a unit square.

E. Specht, The Best Known Packings of Equal Circles in a Square

Cf. A189889, A084616, A084617.

a(2k) = A084617(k). - Jon E. Schoenfield, Jun 09 2019

A343262 a(n) is the number of edges of a regular polygon P with the property that packing n nonoverlapping equal circles inside P, arranged in a configuration with dihedral symmetry D_{2m} with m >= 3, maximizes the packing density.

Original entry on oeis.org

3, 4, 5, 3, 6, 7, 4, 3, 5, 6, 6, 7, 3, 4, 4, 6, 6, 4, 3

Offset: 3

Ya-Ping Lu, Apr 09 2021

Numbers of dihedral symmetries D_{2m} (m >= 3) that n nonoverlapping equal circles possess are given in A343005. The regular polygon is a circle for n=1 and a square for n=2. However, as the symmetry types, O(2) for one circle and D_{4} for two circles, are not D_{2m} with m >= 3, the index of the sequence starts at n = 3.

It can be shown that a(n) <= n and a(n) = k*m/2, where m is the order of a dihedral symmetry of n-circle packing configurations and k is a positive integer.

			For n=3, 3-circle configurations possess one dihedral symmetry D_{6}, or m = 3. Since a(n) must be <= 3 and also a multiple of m, a(n) = 3.
For n = 16, 16-circle configurations have 6 D_{2m} symmetries with m >= 3.
Packing densities are for
m = 16: Pi/(2+2*csc(Pi/8)) = 0.43474+,
m = 15: (8*Pi/15)/(1+csc(2*Pi/15)) = 0.48445+,
m =  8: 4*sqrt(2)*Pi/(1+sqrt(2)+sqrt(3)+sqrt(4-2*sqrt(2)))^2 = 0.65004+,
m =  5: (16*Pi/5)*(7-3*sqrt(5))/sqrt(10+2*sqrt(5)) = 0.77110+,
m =  4: Pi/4 = 0.78539+,
m =  3: 8*Pi/(12+13*sqrt(3)) = 0.72813+.
The highest packing density is achieved at m = 4, or a(16) = 4.
Symmetry type (S) of n-circle configuration giving the highest packing density and the corresponding number of edges (N) of the regular polygon and packing density are given below. The packing configurations are illustrated in the Links.
   n       S      N      Packing density
------  --------  --  -------------------------------------------------------------
   3      D_{6}   3   Pi/(2+4/sqrt(3))                                   = 0.72900+
  4,9,16  D_{8}   4   Pi/4                                               = 0.78539+
   5      D_{10}  5   Pi/(2+8/sqrt(10+2*sqrt(5)))                        = 0.76569+
   6      D_{6}   3   6*Pi/(12+7*sqrt(3))                                = 0.78134+
   7      D_{12}  6   7*Pi/(12+8*sqrt(3))                                = 0.85051+
   8      D_{14}  7   4*Pi/(7+7/sin(2*Pi/7))                             = 0.78769+
  10      D_{6}   3   5*Pi/(9+6*sqrt(3))                                 = 0.81001+
  11      D_{10}  5   (22*Pi/25)/sqrt(10+2*sqrt(5))                      = 0.72671+
  12      D_{6}   6   6*Pi/(12+7*sqrt(3))                                = 0.78134+
  13      D_{12}  6   13*sqrt(3)*Pi/96                                   = 0.73685+
  14      D_{14}  7   4*Pi/(sin(2*Pi/7)*(sqrt(3)+cot(Pi/7)+sec(Pi/7))^2) = 0.66440+
  15      D_{6}   3   15*Pi/(24+19*sqrt(3))                              = 0.82805+
  17      D_{8}   4   (17*Pi/4)/(7+3*sqrt(2)+3*sqrt(3)+sqrt(6))          = 0.70688+
  18      D_{12}  6   9*Pi/(12+13*sqrt(3))                               = 0.81915+
  19      D_{12}  6   19*Pi/(24+26*sqrt(3))                              = 0.86465+
  20      D_{8}   4   20*Pi/(2+sqrt(2)+2*sqrt(3)+sqrt(6))^2              = 0.72213+
  21      D_{6}   3   21*Pi/(30+28*sqrt(3))                              = 0.84045+

Erich Friedman, Packing Equal Copies
Ya-Ping Lu, Illustration of packing configurations
Eckard Specht, Packomania, Packings of equal and unequal circles in fixed-sized containers with maximum packing density

Cf. A023393, A051657, A084616, A084617, A084618, A084644, A133587, A227405, A247397, A253570, A257594, A269110, A308578, A337019, A337020, A343005.

A342843 a(n) is the number of edges of the regular polygon such that packing n nonoverlapping equal circles inside the regular polygon gives the highest packing density. a(n) = 0 if such a regular polygon is a circle.

Original entry on oeis.org

0, 4, 3, 4, 5, 3, 6, 7, 4, 3, 9, 6, 10, 6, 3, 4

Offset: 1

Ya-Ping Lu, Apr 12 2021

Terms for n = 11, 12, 13 and 14 are conjectured values supported by numerical results (see Packomania in the links).
It can be shown that a(n) <= n for n >= 3. As n increases, terms of values other than 3 and 6 will eventually disappear. For example, the packing density of triangular packing of more than 121 circles inside an equilateral triangle, or hexagonal packing of more than 552 circles inside a regular hexagon, is higher than that of square packing inside a square. Thus, for n > 121, the sequence does not have any terms with a(n) = 4.
Conjecture: As n tends to infinity, a(n) takes the value of 3 or 6 and the packing density approaches sqrt(3)*Pi/6.

			a(1) = 0. The maximum packing density for packing 1 circle in regular m-gon is (Pi/m)*cot(Pi/m), which is an increasing function of m. Highest packing density of 1 is achieved as m tends to infinity and the regular n-gon becomes a circle.
a(2) = 4. The maximum packing density for packing 2 circles in regular polygon with odd number of edges m >= 3 is 4*Pi/(m*sin(2*Pi/m))/(sec(Pi/(2*m))+sec(Pi/m))^2, which is smaller than the packing density in regular polygon with even number of edges m >= 4, 4*Pi/(m*sin(2*Pi/m))/(1+sec(Pi/m))^2, which is a decreasing function of m with a maximum of Pi/(3+2*sqrt(2)) at m = 4.
Symmetry type (S) of the n-circle configuration achieving the highest packing density and the corresponding number of edges (N) of the regular polygon and packing density for n up to 16 are listed below.
  n        S     N    Packing density
------  ------  ---   ---------------------------------------------------------
1       O(2)    oo                                                     1
2       D_{4}    4    Pi/(3+2*sqrt(2))                               = 0.53901+
3       D_{6}    3    (Pi/2)/(1+2/sqrt(3))                           = 0.72900+
4,9,16  D_{8}    4    Pi/4                                           = 0.78539+
5       D_{10}   5    (Pi/2)/(1+4/sqrt(10+2*sqrt(5)))                = 0.76569+
6       D_{6}    3    6*Pi/(12+7*sqrt(3))                            = 0.78134+
7       D_{12}   6    7*Pi/(12+8*sqrt(3))                            = 0.85051+
8       D_{14}   7    (4*Pi/7)/(1+1/sin(2*Pi/7))                     = 0.78769+
10      D_{6}    3    (5*Pi/3)/(3+2*sqrt(3))                         = 0.81001+
11      D_{2}    9    (11*Pi/18)/(1+csc(2*Pi/9))                     = 0.75120+
12      D_{6}    6    6*Pi/(12+7*sqrt(3))                            = 0.78134+
13      D_{2}   10    (13*Pi/20)/(1+sqrt(50+10*sqrt(5))/5)           = 0.75594+
14      D_{4}    6    (49*Pi/2)/(21+20*sqrt(3)+6*sqrt(7)+6*sqrt(21)) = 0.77737+
15      D_{6}    3    15*Pi/(24+19*sqrt(3))                          = 0.82805+

Erich Friedman, Packing Equal Copies
Eckard Specht, Packomania, Packings of equal and unequal circles in fixed-sized containers with maximum packing density

Cf. A023393, A051657, A084616, A084617, A084618, A084644, A133587, A227405, A247397, A253570, A257594, A269110, A308578, A337019, A337020, A343005, A343262.

cp's OEIS Frontend

A084617 Maximum number of circles with diameter 1 that can be packed in a square of side length n.

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A084618 Maximum number of circles of area 1 that can be packed in a circle of area n.

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A084824 Maximum number of spheres of diameter one that can be packed in a cube of volume n (i.e., with edge length n^(1/3)).

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A253570 Maximum number of circles of radius 1 that can be packed into a regular n-gon with side length 2 (conjectured).

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A337020 Maximum number of circles with unit area that can be packed into a square with an area of n.

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A308578 Maximum number of non-overlapping circles of radius 1/n that can be placed inside a unit square.

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A343262 a(n) is the number of edges of a regular polygon P with the property that packing n nonoverlapping equal circles inside P, arranged in a configuration with dihedral symmetry D_{2m} with m >= 3, maximizes the packing density.

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A342843 a(n) is the number of edges of the regular polygon such that packing n nonoverlapping equal circles inside the regular polygon gives the highest packing density. a(n) = 0 if such a regular polygon is a circle.

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