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%I #39 Nov 10 2020 13:03:24
%S 0,1,1,2,3,4,4,5,5,6,8,9,9,9,10,11,12,13,13,15,16,16,16,18,18,20,20,
%T 21,22,22,24,25,25,25,27,27,28,30,30
%N Maximum number of circles with unit area that can be packed into a square with an area of n.
%C Terms beyond a(n) = 30 (n = 38 & 39) except a(n) = 36 are conjectures supported by numerical results (see Packomania in the links) and terms for n from 40 through 70 are: 31, 31, 33, 33, 34, 35, 36, 36, 38, 39, 39, 40, 41, 42, 42, 43, 44, 44, 46, 46, 47, 48, 49, 50, 52, 52, 53, 53, 54, 55, 56.
%C The packing density, a(n)/n, approaches sqrt(3)*Pi/6 as n tends to infinity.
%C References for the known optimal packings are given in Table 1.2 on page 10 and the bibliography on pages 219-225 of the book by Szabo et al. (see References).
%D P. G. Szabo, M. Cs. Markot, T. Csendes, E. Specht, L. G. Casado, and I. Garcia, New Approaches to Circle Packing in a Square, Springer, 2007.
%H R. L. Graham and B. D. Lubachevsky, <a href="http://arxiv.org/abs/math/0406394">Repeated Patterns of Dense Packings of Equal Disks in a Square</a>, Electronic Journal of Combinatorics 3 (1996), #R16; arXiv:math/0406394 [math.MG], 2004.
%H Eckard Specht, editor, <a href="http://www.packomania.com">Packomania</a>, Section 1: Packings of equal and unequal circles in fixed-sized containers with maximum packing density.
%Y Cf. A084616, A084617, A084644, A093766, A337019.
%K nonn,more
%O 1,4
%A _Ya-Ping Lu_, Nov 06 2020
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