cp's OEIS Frontend

A084616 Maximum number of circles of diameter 1 that can be packed in a square of area n (i.e., with side length n^(1/2)).

%I A084616 #19 Sep 17 2024 08:44:48
%S A084616 1,1,2,4,4,5,5,6,9,9,9,10,12,13,14,16,16,16,18,19,20,21,22,23,25,25,
%T A084616 26,27,28,30,30,31,33,33,34,36,36,39,39,40,41,42,43,44,45,46,47,48,49,
%U A084616 52,52,53,53,55,56,57,58,59,59,61,62,63,65,68,68,68,69,69,70,72,73,74,74
%N A084616 Maximum number of circles of diameter 1 that can be packed in a square of area n (i.e., with side length n^(1/2)).
%C A084616 Most sequence terms beyond n=20 are only conjectures supported by comprehensive numerical results. No proof is available for the following observations: n=30 is the first case where a square of area < n (29.74921576) is sufficient to cover n circles. The first case where more than n circles can be covered occurs for n=38. The required area to cover 39 circles is 37.76050335. n=59 is the last case where a square of area n does not suffice to cover n+1 circles (60 circles require square area 59.11626524).
%H A084616 Mihály Csaba Markót, <a href="https://doi.org/10.1007/s10898-021-01086-z">Improved interval methods for solving circle packing problems in the unit square</a>. J Glob Optim 81, 773-803 (2021).
%H A084616 Hugo Pfoertner, <a href="https://www.randomwalk.de/sequences/a084616.pdf">Minimum area of square needed to cover n circles of diameter 1</a>.
%H A084616 E. Specht, <a href="http://hydra.nat.uni-magdeburg.de/packing/csq/csq.html">The best known packings of equal circles in the unit square</a>.
%H A084616 P. G. Szabó et al., <a href="http://dx.doi.org/10.1007/978-0-387-45676-8">New Approaches to Circle Packing in a Square</a>, Vol. 6 in Optimization and Its Applications, Springer 2007.
%e A084616 a(2)=1 because a square of side length sqrt(2)=1.414... is not large enough to cover two circles of diameter 1 (the required side length would be 1+sqrt(2)/2=1.707...).
%e A084616 a(38)=39 because 39 circles fit into a square of area 38.
%Y A084616 Cf. A051657, A084617, A084618.
%K A084616 nonn
%O A084616 1,3
%A A084616 _Hugo Pfoertner_, Jun 01 2003