cp's OEIS Frontend

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

%I A121346 #25 Jun 12 2025 12:58:31
%S A121346 2,11,31,68,124,205,316,460,642,866,1138,1461,1839,2278,2781,3354,
%T A121346 4000,4724,5531,6424,7409,8490,9671,10956,12351,13859,15485,17234,
%U A121346 19110,21116,23259,25542,27969,30546,33276,36164,39215,42432,45821,49385
%N A121346 Conjectured lower bound for the number of spheres of radius 1 that can be packed in a sphere of radius n.
%C A121346 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.
%C A121346 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
%H A121346 Hugo Pfoertner, <a href="/A121346/b121346.txt">Table of n, a(n) for n = 2..10000</a>
%H A121346 Sen Bai, X. Bai, X. Che, and X. Wei, <a href="https://doi.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. 1 2016), pp. 2023-2033.
%H A121346 David W. Cantrell, <a href="https://groups.google.com/g/sci.math/c/Q9OMtGmXNbg/m/a_AnL8wYel8J">Packing many equal small spheres into a large sphere</a>, post in newsgroup sci.math, May 29 2006.
%H A121346 WenQi Huang and Liang Yu, <a href="https://doi.ieeecomputersociety.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.
%F A121346 a(n) = floor(K*(1 - 2*d)/d^3 + 1/(2*d^2)), where d=1/n and K = Pi/(3*sqrt(2)) (A093825).
%t A121346 A121346[n_] := Floor[n^2*(Pi*(n - 2)*Sqrt[2] + 3)/6];
%t A121346 Array[A121346, 50, 2] (* _Paolo Xausa_, Jun 12 2025 *)
%Y A121346 Cf. A084828, A093825.
%K A121346 nonn
%O A121346 2,1
%A A121346 _Hugo Pfoertner_, Jul 22 2006