A121346 Conjectured lower bound for the number of spheres of radius 1 that can be packed in a sphere of radius n.

2, 11, 31, 68, 124, 205, 316, 460, 642, 866, 1138, 1461, 1839, 2278, 2781, 3354, 4000, 4724, 5531, 6424, 7409, 8490, 9671, 10956, 12351, 13859, 15485, 17234, 19110, 21116, 23259, 25542, 27969, 30546, 33276, 36164, 39215, 42432, 45821, 49385

Offset: 2

Author

Hugo Pfoertner, Jul 22 2006

Keywords

• nonn

Comments

The formula was given by David W. Cantrell in a thread "Packing many equal small spheres into a larger sphere" in the newsgroup sci.math on May 29 2006.

Cantrell's formula can be expressed quite accurately using an easy-to-remember rule of thumb: a(n) = n^2*((3/4)*n - 1). To be even more precise, subtract 1%, i.e., multiply by a factor of 0.99. - Hugo Pfoertner, Jun 12 2025

Links

• Hugo Pfoertner, Table of n, a(n) for n = 2..10000
• Sen Bai, X. Bai, X. Che, and X. Wei, Maximal Independent Sets in Heterogeneous Wireless Ad Hoc Networks, IEEE Transactions on Mobile Computing (Volume: 15, Issue: 8, Aug. 1 2016), pp. 2023-2033.
• David W. Cantrell, Packing many equal small spheres into a large sphere, post in newsgroup sci.math, May 29 2006.
• WenQi Huang and Liang Yu, A Quasi Physical Method for the Equal Sphere Packing Problem, in 2011 IEEE 10th International Conference on Trust, Security and Privacy in Computing and Communications.

Crossrefs

Cf. A084828, A093825.

Programs

Mathematica

A121346[n_] := Floor[n^2*(Pi*(n - 2)*Sqrt[2] + 3)/6];
Array[A121346, 50, 2] (* Paolo Xausa, Jun 12 2025 *)

Formula

a(n) = floor(K*(1 - 2*d)/d^3 + 1/(2*d^2)), where d=1/n and K = Pi/(3*sqrt(2)) (A093825).