This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A343262 #25 May 20 2021 12:48:40
%S A343262 3,4,5,3,6,7,4,3,5,6,6,7,3,4,4,6,6,4,3
%N 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.
%C A343262 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.
%C A343262 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.
%H A343262 Erich Friedman, <a href="https://erich-friedman.github.io/packing/">Packing Equal Copies</a>
%H A343262 Ya-Ping Lu, <a href="/A343262/a343262_1.pdf">Illustration of packing configurations</a>
%H A343262 Eckard Specht, Packomania, <a href="http://www.packomania.com">Packings of equal and unequal circles in fixed-sized containers with maximum packing density</a>
%e A343262 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.
%e A343262 For n = 16, 16-circle configurations have 6 D_{2m} symmetries with m >= 3.
%e A343262 Packing densities are for
%e A343262 m = 16: Pi/(2+2*csc(Pi/8)) = 0.43474+,
%e A343262 m = 15: (8*Pi/15)/(1+csc(2*Pi/15)) = 0.48445+,
%e A343262 m = 8: 4*sqrt(2)*Pi/(1+sqrt(2)+sqrt(3)+sqrt(4-2*sqrt(2)))^2 = 0.65004+,
%e A343262 m = 5: (16*Pi/5)*(7-3*sqrt(5))/sqrt(10+2*sqrt(5)) = 0.77110+,
%e A343262 m = 4: Pi/4 = 0.78539+,
%e A343262 m = 3: 8*Pi/(12+13*sqrt(3)) = 0.72813+.
%e A343262 The highest packing density is achieved at m = 4, or a(16) = 4.
%e A343262 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.
%e A343262 n S N Packing density
%e A343262 ------ -------- -- -------------------------------------------------------------
%e A343262 3 D_{6} 3 Pi/(2+4/sqrt(3)) = 0.72900+
%e A343262 4,9,16 D_{8} 4 Pi/4 = 0.78539+
%e A343262 5 D_{10} 5 Pi/(2+8/sqrt(10+2*sqrt(5))) = 0.76569+
%e A343262 6 D_{6} 3 6*Pi/(12+7*sqrt(3)) = 0.78134+
%e A343262 7 D_{12} 6 7*Pi/(12+8*sqrt(3)) = 0.85051+
%e A343262 8 D_{14} 7 4*Pi/(7+7/sin(2*Pi/7)) = 0.78769+
%e A343262 10 D_{6} 3 5*Pi/(9+6*sqrt(3)) = 0.81001+
%e A343262 11 D_{10} 5 (22*Pi/25)/sqrt(10+2*sqrt(5)) = 0.72671+
%e A343262 12 D_{6} 6 6*Pi/(12+7*sqrt(3)) = 0.78134+
%e A343262 13 D_{12} 6 13*sqrt(3)*Pi/96 = 0.73685+
%e A343262 14 D_{14} 7 4*Pi/(sin(2*Pi/7)*(sqrt(3)+cot(Pi/7)+sec(Pi/7))^2) = 0.66440+
%e A343262 15 D_{6} 3 15*Pi/(24+19*sqrt(3)) = 0.82805+
%e A343262 17 D_{8} 4 (17*Pi/4)/(7+3*sqrt(2)+3*sqrt(3)+sqrt(6)) = 0.70688+
%e A343262 18 D_{12} 6 9*Pi/(12+13*sqrt(3)) = 0.81915+
%e A343262 19 D_{12} 6 19*Pi/(24+26*sqrt(3)) = 0.86465+
%e A343262 20 D_{8} 4 20*Pi/(2+sqrt(2)+2*sqrt(3)+sqrt(6))^2 = 0.72213+
%e A343262 21 D_{6} 3 21*Pi/(30+28*sqrt(3)) = 0.84045+
%Y A343262 Cf. A023393, A051657, A084616, A084617, A084618, A084644, A133587, A227405, A247397, A253570, A257594, A269110, A308578, A337019, A337020, A343005.
%K A343262 nonn,more
%O A343262 3,1
%A A343262 _Ya-Ping Lu_, Apr 09 2021