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%I #26 Apr 15 2021 01:11:58
%S 0,4,3,4,5,3,6,7,4,3,9,6,10,6,3,4
%N 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.
%C Terms for n = 11, 12, 13 and 14 are conjectured values supported by numerical results (see Packomania in the links).
%C 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.
%C Conjecture: As n tends to infinity, a(n) takes the value of 3 or 6 and the packing density approaches sqrt(3)*Pi/6.
%H Erich Friedman, <a href="https://erich-friedman.github.io/packing/">Packing Equal Copies</a>
%H 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 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.
%e 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.
%e 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.
%e n S N Packing density
%e ------ ------ --- ---------------------------------------------------------
%e 1 O(2) oo 1
%e 2 D_{4} 4 Pi/(3+2*sqrt(2)) = 0.53901+
%e 3 D_{6} 3 (Pi/2)/(1+2/sqrt(3)) = 0.72900+
%e 4,9,16 D_{8} 4 Pi/4 = 0.78539+
%e 5 D_{10} 5 (Pi/2)/(1+4/sqrt(10+2*sqrt(5))) = 0.76569+
%e 6 D_{6} 3 6*Pi/(12+7*sqrt(3)) = 0.78134+
%e 7 D_{12} 6 7*Pi/(12+8*sqrt(3)) = 0.85051+
%e 8 D_{14} 7 (4*Pi/7)/(1+1/sin(2*Pi/7)) = 0.78769+
%e 10 D_{6} 3 (5*Pi/3)/(3+2*sqrt(3)) = 0.81001+
%e 11 D_{2} 9 (11*Pi/18)/(1+csc(2*Pi/9)) = 0.75120+
%e 12 D_{6} 6 6*Pi/(12+7*sqrt(3)) = 0.78134+
%e 13 D_{2} 10 (13*Pi/20)/(1+sqrt(50+10*sqrt(5))/5) = 0.75594+
%e 14 D_{4} 6 (49*Pi/2)/(21+20*sqrt(3)+6*sqrt(7)+6*sqrt(21)) = 0.77737+
%e 15 D_{6} 3 15*Pi/(24+19*sqrt(3)) = 0.82805+
%Y Cf. A023393, A051657, A084616, A084617, A084618, A084644, A133587, A227405, A247397, A253570, A257594, A269110, A308578, A337019, A337020, A343005, A343262.
%K more,nonn
%O 1,2
%A _Ya-Ping Lu_, Apr 12 2021
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