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

1, 2, 6, 11, 18, 26, 37, 49, 63, 79, 97, 116, 138, 161, 186, 213, 241, 272, 304, 338, 374, 412, 451, 492, 535, 580, 627, 676, 726, 778, 832, 888, 946, 1005, 1066, 1130, 1194, 1261, 1330, 1400, 1472, 1546, 1622, 1699, 1779, 1860, 1943, 2028, 2115, 2203, 2293, 2385

Offset: 1

Author

Hugo Pfoertner, Dec 07 2011

Keywords

• nonn

Comments

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.

Links

• Hugo Pfoertner, Table of n, a(n) for n = 1..1051
• David W. Cantrell, A Conjectured Upper Bound for r. Posting in thread "Packing unit circles in circle: new results" in newsgroup sci.math, Dec 6 2008.
• Hugo Pfoertner, Comparison of best known packings against Cantrell's bound. (2014)
• E. Specht, The best known packings of equal circles in a circle

Crossrefs

Cf. A023393 (best known packings).

Programs

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

Formula

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)).