OFFSET
1,3
COMMENTS
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).
LINKS
Mihály Csaba Markót, Improved interval methods for solving circle packing problems in the unit square. J Glob Optim 81, 773-803 (2021).
Hugo Pfoertner, Minimum area of square needed to cover n circles of diameter 1.
P. G. Szabó et al., New Approaches to Circle Packing in a Square, Vol. 6 in Optimization and Its Applications, Springer 2007.
EXAMPLE
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...).
a(38)=39 because 39 circles fit into a square of area 38.
CROSSREFS
KEYWORD
nonn
AUTHOR
Hugo Pfoertner, Jun 01 2003
STATUS
approved