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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 Table of n, a(n) for n=3..21. 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 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 Cf. A023393, A051657, A084616, A084617, A084618, A084644, A133587, A227405, A247397, A253570, A257594, A269110, A308578, A337019, A337020, A343005. Sequence in context: A121890 A330740 A178231 * A298734 A137926 A218342 Adjacent sequences: A343259 A343260 A343261 * A343263 A343264 A343265 KEYWORD nonn,more AUTHOR Ya-Ping Lu, Apr 09 2021 STATUS approved

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