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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.

%I A342843 #26 Apr 15 2021 01:11:58
%S A342843 0,4,3,4,5,3,6,7,4,3,9,6,10,6,3,4
%N 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.
%C A342843 Terms for n = 11, 12, 13 and 14 are conjectured values supported by numerical results (see Packomania in the links).
%C A342843 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 A342843 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 A342843 Erich Friedman, <a href="https://erich-friedman.github.io/packing/">Packing Equal Copies</a>
%H A342843 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 A342843 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 A342843 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 A342843 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 A342843   n        S     N    Packing density
%e A342843 ------  ------  ---   ---------------------------------------------------------
%e A342843 1       O(2)    oo                                                     1
%e A342843 2       D_{4}    4    Pi/(3+2*sqrt(2))                               = 0.53901+
%e A342843 3       D_{6}    3    (Pi/2)/(1+2/sqrt(3))                           = 0.72900+
%e A342843 4,9,16  D_{8}    4    Pi/4                                           = 0.78539+
%e A342843 5       D_{10}   5    (Pi/2)/(1+4/sqrt(10+2*sqrt(5)))                = 0.76569+
%e A342843 6       D_{6}    3    6*Pi/(12+7*sqrt(3))                            = 0.78134+
%e A342843 7       D_{12}   6    7*Pi/(12+8*sqrt(3))                            = 0.85051+
%e A342843 8       D_{14}   7    (4*Pi/7)/(1+1/sin(2*Pi/7))                     = 0.78769+
%e A342843 10      D_{6}    3    (5*Pi/3)/(3+2*sqrt(3))                         = 0.81001+
%e A342843 11      D_{2}    9    (11*Pi/18)/(1+csc(2*Pi/9))                     = 0.75120+
%e A342843 12      D_{6}    6    6*Pi/(12+7*sqrt(3))                            = 0.78134+
%e A342843 13      D_{2}   10    (13*Pi/20)/(1+sqrt(50+10*sqrt(5))/5)           = 0.75594+
%e A342843 14      D_{4}    6    (49*Pi/2)/(21+20*sqrt(3)+6*sqrt(7)+6*sqrt(21)) = 0.77737+
%e A342843 15      D_{6}    3    15*Pi/(24+19*sqrt(3))                          = 0.82805+
%Y A342843 Cf. A023393, A051657, A084616, A084617, A084618, A084644, A133587, A227405, A247397, A253570, A257594, A269110, A308578, A337019, A337020, A343005, A343262.
%K A342843 more,nonn
%O A342843 1,2
%A A342843 _Ya-Ping Lu_, Apr 12 2021