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)).

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, 26, 27, 28, 30, 30, 31, 33, 33, 34, 36, 36, 39, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 52, 52, 53, 53, 55, 56, 57, 58, 59, 59, 61, 62, 63, 65, 68, 68, 68, 69, 69, 70, 72, 73, 74, 74

Offset: 1

Author

Hugo Pfoertner, Jun 01 2003

Keywords

• nonn

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).

Examples

			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.
		

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.
• E. Specht, The best known packings of equal circles in the unit square.
• P. G. Szabó et al., New Approaches to Circle Packing in a Square, Vol. 6 in Optimization and Its Applications, Springer 2007.

Crossrefs

Cf. A051657, A084617, A084618.