A337020 Maximum number of circles with unit area that can be packed into a square with an area of n.

0, 1, 1, 2, 3, 4, 4, 5, 5, 6, 8, 9, 9, 9, 10, 11, 12, 13, 13, 15, 16, 16, 16, 18, 18, 20, 20, 21, 22, 22, 24, 25, 25, 25, 27, 27, 28, 30, 30

Offset: 1

Author

Ya-Ping Lu, Nov 06 2020

Keywords

• nonn
• more

Comments

Terms beyond a(n) = 30 (n = 38 & 39) except a(n) = 36 are conjectures supported by numerical results (see Packomania in the links) and terms for n from 40 through 70 are: 31, 31, 33, 33, 34, 35, 36, 36, 38, 39, 39, 40, 41, 42, 42, 43, 44, 44, 46, 46, 47, 48, 49, 50, 52, 52, 53, 53, 54, 55, 56.

The packing density, a(n)/n, approaches sqrt(3)*Pi/6 as n tends to infinity.

References for the known optimal packings are given in Table 1.2 on page 10 and the bibliography on pages 219-225 of the book by Szabo et al. (see References).

References

• P. G. Szabo, M. Cs. Markot, T. Csendes, E. Specht, L. G. Casado, and I. Garcia, New Approaches to Circle Packing in a Square, Springer, 2007.

Links

• R. L. Graham and B. D. Lubachevsky, Repeated Patterns of Dense Packings of Equal Disks in a Square, Electronic Journal of Combinatorics 3 (1996), #R16; arXiv:math/0406394 [math.MG], 2004.
• Eckard Specht, editor, Packomania, Section 1: Packings of equal and unequal circles in fixed-sized containers with maximum packing density.

Crossrefs

Cf. A084616, A084617, A084644, A093766, A337019.