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A201993 Conjectured lower bound for the number of circles of radius 1 that can be packed into a circle of radius n.

%I A201993 #16 Aug 01 2024 14:12:11
%S A201993 1,2,6,11,18,26,37,49,63,79,97,116,138,161,186,213,241,272,304,338,
%T A201993 374,412,451,492,535,580,627,676,726,778,832,888,946,1005,1066,1130,
%U A201993 1194,1261,1330,1400,1472,1546,1622,1699,1779,1860,1943,2028,2115,2203,2293,2385
%N A201993 Conjectured lower bound for the number of circles of radius 1 that can be packed into a circle of radius n.
%C A201993 Bound provided by David W. Cantrell in December 2008. It is conjectured that it is possible to find packings such that A023393(n)>=a(n) for all n. Currently (December 2011) the smallest number of circles, for which the bound is not achieved, is 507.
%H A201993 Hugo Pfoertner, <a href="/A201993/b201993.txt">Table of n, a(n) for n = 1..1051</a>
%H A201993 David W. Cantrell, <a href="https://groups.google.com/group/sci.math/msg/acdc41a82009833e">A Conjectured Upper Bound for r.</a> Posting in thread "Packing unit circles in circle: new results" in newsgroup sci.math, Dec 6 2008.
%H A201993 Hugo Pfoertner, <a href="/A201993/a201993.pdf">Comparison of best known packings against Cantrell's bound</a>. (2014)
%H A201993 E. Specht, <a href="http://hydra.nat.uni-magdeburg.de/packing/cci/cci.html">The best known packings of equal circles in a circle</a>
%F A201993 a(n) = Smallest k, such that 1 + (sqrt((4*Rho-1)^2 + 16*Rho*(k-1)) - 1) / (4*Rho) >=n with Rho = Pi/(2*sqrt(3)).
%o A201993 (PARI) for(k=2,53,my(rho=Pi/(2*sqrt(3)),N(R)=rho*R*(R-2)+R/2+1);print1(ceil(N(k-1)),", ")) \\ _Hugo Pfoertner_, Aug 02 2019
%Y A201993 Cf. A023393 (best known packings).
%K A201993 nonn
%O A201993 1,2
%A A201993 _Hugo Pfoertner_, Dec 07 2011