OFFSET
2,1
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.
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).
MATHEMATICA
A121346[n_] := Floor[n^2*(Pi*(n - 2)*Sqrt[2] + 3)/6];
Array[A121346, 50, 2] (* Paolo Xausa, Jun 12 2025 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Hugo Pfoertner, Jul 22 2006
STATUS
approved
