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

1,4

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

Table of n, a(n) for n=1..39.

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.

Sequence in context: A096336 A358062 A081609 * A351519 A046700 A216411

Adjacent sequences: A337017 A337018 A337019 * A337021 A337022 A337023

Keyword

nonn,more

Author

Ya-Ping Lu, Nov 06 2020

Status

approved