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Maximum number of spheres of radius one that can be packed in a sphere of radius n.
5

%I #36 Jan 22 2024 14:42:09

%S 1,2,13,32,68

%N Maximum number of spheres of radius one that can be packed in a sphere of radius n.

%C a(4) and a(5) are experimental values. Although A121346(5) claims a lower bound of a(5)=68, it is conjectured from extensive numerical search that this value is unachievable and therefore a(5)=67.

%C The conjecture a(5)=67 was proved wrong by Yu Liang, who found an arrangement of 68 spheres of radius 1 fitting into a sphere of radius 5.

%C Lower bounds for the next terms are a(6)>=122 and a(7)>=198. See E. Specht's webpage for latest data. - _Hugo Pfoertner_, Jan 22 2024

%H Sen Bai, X. Bai, X. Che, and X. Wei, <a href="https://doi.ieeecomputersociety.org/10.1109/TMC.2015.2508805">Maximal Independent Sets in Heterogeneous Wireless Ad Hoc Networks</a>, IEEE Transactions on Mobile Computing (Volume: 15, Issue: 8, Aug 01 2016), pp. 2023-2033.

%H Dave Boll, <a href="http://web.archive.org/web/20121213221349/https://home.comcast.net/~davejanelle/packing.html">Optimal Packing of Circles and Spheres</a>.

%H Sunil K. Chebolu, <a href="https://arxiv.org/abs/2006.00603">Packing Moons Inside the Earth</a>, arXiv:2006.00603 [physics.pop-ph], 2020.

%H WenQi Huang and Liang Yu, <a href="https://doi.org/10.1109/TrustCom.2011.233">A Quasi Physical Method for the Equal Sphere Packing Problem</a>, in 2011 IEEE 10th International Conference on Trust, Security and Privacy in Computing and Communications.

%H WenQi Huang and Liang Yu, <a href="http://arxiv.org/abs/1202.4149">Serial Symmetrical Relocation Algorithm for the Equal Sphere Packing Problem</a>, arXiv preprint arXiv:1202.4149 [cs.DM], 2012. - From _N. J. A. Sloane_, Jun 14 2012

%H Hugo Pfoertner, <a href="https://www.randomwalk.de/sequences/a084827.txt">Numerical results for best packing of spheres in sphere</a>.

%H Hugo Pfoertner, <a href="https://www.randomwalk.de/sphere/insphr/spheresinsphr.html">Densest Packing of Spheres in a Sphere. Java visualization.</a>

%H Eckhard Specht, <a href="http://hydra.nat.uni-magdeburg.de/packing/ssp/ssp.html">The best known packings of equal spheres in a sphere</a>.

%H Yu Liang, <a href="/A084828/a084828.txt">Coordinates of sphere centers of 68 spheres of radius 0.20000222, fitting into a container of radius 1.</a> Private communication, Aug 22 2011.

%Y Cf. A121346 (conjectured lower bounds), A084827, A084829, A084825.

%Y Cf. A023393 (2D).

%K hard,more,nonn

%O 1,2

%A _Hugo Pfoertner_, Jun 12 2003

%E Comment and links edited, a(5) from _Hugo Pfoertner_, Jun 23 2011

%E a(5) corrected, based on private communication from Yu Liang, by _Hugo Pfoertner_, Aug 24 2011