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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. 2
3, 4, 5, 3, 6, 7, 4, 3, 5, 6, 6, 7, 3, 4, 4, 6, 6, 4, 3 (list; graph; refs; listen; history; text; internal format)
OFFSET
3,1
COMMENTS
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.
LINKS
Erich Friedman, Packing Equal Copies
EXAMPLE
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+
CROSSREFS
Sequence in context: A121890 A330740 A178231 * A298734 A137926 A218342
KEYWORD
nonn,more
AUTHOR
Ya-Ping Lu, Apr 09 2021
STATUS
approved

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Last modified May 23 01:37 EDT 2024. Contains 372758 sequences. (Running on oeis4.)