1,3

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 were a square of area < n (29.74921576) is sufficient to cover n circles. The first case were 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).

List of references given by E. Specht and P. Szabo, see corresponding links.

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. Szabo, Equal circles packing in a square.

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.

Cf. A051657, A084617, A084618.

Sequence in context: A036437 A053306 A108422 this_sequence A071193 A071192 A100921

Adjacent sequences: A084613 A084614 A084615 this_sequence A084617 A084618 A084619

nonn

Hugo Pfoertner (hugo(AT)pfoertner.org), Jun 01 2003