A253570 Maximum number of circles of radius 1 that can be packed into a regular n-gon with side length 2 (conjectured).

0, 1, 1, 1, 3, 4, 5, 7, 8, 9

Offset: 3

Author

Felix Fröhlich, Jan 03 2015

Keywords

• nonn
• hard
• more

Comments

The values were obtained by constructing the circle arrangements in a vector graphics program and have not been proved to be correct.

From David Consiglio, Jr., Jan 09 2015: (Start)

As n increases, the n-gon more and more closely approximates a circle. As a result, the lower bound (which is highly likely to be the correct term for larger and larger n) is the number of circles that can be packed into an inscribed circle, the radius of which is given by the expression cot(Pi/n). Look up this radius in column 3 at www.packomania.com to find the lower bound of a(n).

A rough upper bound would be the closest packing of circles into the area of the n-gon (formula below). A better upper bound is likely possible.

See file for lower and upper bounds through a(20). The lower bounds have been proved for a(3) through a(13).

(End)

Links

• David Consiglio, Jr., Lower and Upper Bounds
• Felix Fröhlich, Illustration of circle arrangements associated with a(3)-a(12)
• E. Specht, The best known packings of equal circles in a circle (complete up to N = 2600), Packomania.

Crossrefs

Cf. A023393, A051657, A084616.

Formula

Upper bound = floor(n/(2*sqrt(3)*tan(Pi/n))).