A084618 -id:A084618 - cp's OEIS frontend

A023393 Maximal number of circles of radius 1 that can be packed in a circle of radius n.

Original entry on oeis.org

0, 1, 2, 7, 11, 19, 27, 38, 50, 64, 80, 98, 118

Offset: 0

David W. Wilson

The terms for n>5 are only conjectures supported by extensive computations.

R. L. Graham and B. D. Lubachevsky, Dense packings of 3k(k+1)+1 equal disks in a circle for k = 1, 2, 3, 4 and 5, Proc. First Int. Conf. "Computing and Combinatorics" COCOON'95, Springer Lecture Notes in Computer Science 959 (1996), 303-312.
For list of references given by E. Specht, see the Specht link.

Claudio Chaib, GeometricScene: Circles in Circle packing (A023393 OEIS), Apr 02 2020.
R. L. Graham, B. D. Lubachevsky, K. J. Nurmela, and P. R. J. Östergård, Dense Packings of Congruent Circles in a Circle, Discrete Mathematics 181 (1998), 139-154.
B. D. Lubachevsky and R. L. Graham, Curved Hexagonal Packings of Equal Disks in a Circle, Discrete Comput. Geom. 18 (1997), 179-194.
E. Specht, The best known packings of equal circles in the unit circle

Cf. A084618, A084617, A084644.

Cf. A201993 (conjectured lower bounds for a(n)).

Terms for n>5 from Hugo Pfoertner, Jun 01 2003
Mohammed Bouayoun (mohammed.bouayoun(AT)sanef.com), Feb 18 2004, writes to suggest that the sequence probably continues 138, 161, 187, 213, 242, 272, 304, 337, 373, 413, 451, 495, ...
Edited by N. J. A. Sloane at the suggestion of David W. Wilson, Sep 22 2007
Offset corrected by Jon E. Schoenfield, Oct 12 2008

A084616 Maximum number of circles of diameter 1 that can be packed in a square of area n (i.e., with side length n^(1/2)).

Original entry on oeis.org

1, 1, 2, 4, 4, 5, 5, 6, 9, 9, 9, 10, 12, 13, 14, 16, 16, 16, 18, 19, 20, 21, 22, 23, 25, 25, 26, 27, 28, 30, 30, 31, 33, 33, 34, 36, 36, 39, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 52, 52, 53, 53, 55, 56, 57, 58, 59, 59, 61, 62, 63, 65, 68, 68, 68, 69, 69, 70, 72, 73, 74, 74

Offset: 1

Hugo Pfoertner, Jun 01 2003

nonn

Most sequence terms beyond n=20 are only conjectures supported by comprehensive numerical results. No proof is available for the following observations: n=30 is the first case where a square of area < n (29.74921576) is sufficient to cover n circles. The first case where more than n circles can be covered occurs for n=38. The required area to cover 39 circles is 37.76050335. n=59 is the last case where a square of area n does not suffice to cover n+1 circles (60 circles require square area 59.11626524).

			a(2)=1 because a square of side length sqrt(2)=1.414... is not large enough to cover two circles of diameter 1 (the required side length would be 1+sqrt(2)/2=1.707...).
a(38)=39 because 39 circles fit into a square of area 38.

Mihály Csaba Markót, Improved interval methods for solving circle packing problems in the unit square. J Glob Optim 81, 773-803 (2021).
Hugo Pfoertner, Minimum area of square needed to cover n circles of diameter 1.
E. Specht, The best known packings of equal circles in the unit square.
P. G. Szabó et al., New Approaches to Circle Packing in a Square, Vol. 6 in Optimization and Its Applications, Springer 2007.

Cf. A051657, A084617, A084618.

A084644 Best packings of m>1 equal circles into a larger circle setting a new density record.

Original entry on oeis.org

2, 3, 4, 7, 19, 37, 55, 85, 121, 147, 148, 150, 151, 187, 188, 190, 191, 192, 193, 198, 199, 235, 241, 264, 267, 269, 270, 291, 292, 293, 294, 295, 343, 346, 347, 348, 349, 408, 409, 412, 414, 415, 417, 418, 419, 420, 421, 481, 499, 564, 565, 649, 689, 690, 721

Offset: 1

Hugo Pfoertner, Jun 01 2003

nonn

Sequence terms for n>5 are only conjectures. The arrangement of 37 circles consists of one central circle surrounded by 3 rings of 6,12 and 18 circles. For n=7, 49 of the 55 circles are arranged in a rigid hexagonal lattice with 6 "rattlers" inserted in gaps at the circumference.

			a(4)=7 because the density 0.7777.. of the best packing of 7 circles (1 central circle surrounded by 6 neighbors) exceeds the density 0.68629.. of the packing of 4 circles arranged in a square.

List of references given by E. Specht; see corresponding link.

Robert G. Wilson v, Table of n, a(n) for n = 1..85
Jerry Donovan, Packing Circles in Squares and Circles Page.
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
E. Specht, The sequence of N's that establish density records - provides continuation of the sequence.

Cf. A051657 (density records for circles packed into square), A084618, A023393.

More terms from Robert G. Wilson v, Nov 07 2012

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.

A384851 Decimal expansion of minimal radius of a circle that contains 14 non-overlapping unit disks.

Original entry on oeis.org

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

Offset: 1

Jinyuan Wang, Jun 10 2025

			4.328428554860836681403909367478181...

Dinesh B. Ekanayake and Douglas J. LaFountain, Tight partitions for packing circles in a circle, Italian Journal of Pure and Applied Mathematics, 51 (2024), 115-136.
Michael Goldberg, Packing of 14, 16, 17 and 20 Circles in a Circle, Mathematics Magazine, Vol. 44, No. 3 (May, 1971), pp. 134-139 (provides D/d=4.3284 in Table 1, page 136).

Cf. A084618, A121570, A121598, A121601, A121602, A281115, A282279.

1+sqrt(1+polrootsreal(Pol([707281, -27270266, 472689975, -4930771548, 34449512067, -168736166642, 591369611109, -1498751280720, 2767422383674, -3746579404052, 3734397946902, -2743990597288, 1486108072662, -593729401364, 175537055738, -38557290064, 6295485573, -759438450, 66647843, -4134492, 172311, -4346, 49]))[10])

cp's OEIS Frontend

A023393 Maximal number of circles of radius 1 that can be packed in a circle of radius n.

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A084616 Maximum number of circles of diameter 1 that can be packed in a square of area n (i.e., with side length n^(1/2)).

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A084644 Best packings of m>1 equal circles into a larger circle setting a new density record.

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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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A384851 Decimal expansion of minimal radius of a circle that contains 14 non-overlapping unit disks.

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