

A201993


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


2



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
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OFFSET

1,2


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


FORMULA

a(n) = Smallest k, such that 1 + (sqrt((4*Rho1)^2 + 16*Rho*(k1))  1) / (4*Rho) >=n with Rho = Pi/(2*sqrt(3)).


PROG

(PARI) for(k=2, 53, my(rho=Pi/(2*sqrt(3)), N(R)=rho*R*(R2)+R/2+1); print1(ceil(N(k1)), ", "))  Hugo Pfoertner, Aug 02 2019


CROSSREFS

Cf. A023393 (best known packings).
Sequence in context: A248469 A225386 A037258 * A024521 A194455 A163324
Adjacent sequences: A201990 A201991 A201992 * A201994 A201995 A201996


KEYWORD

nonn


AUTHOR

Hugo Pfoertner, Dec 07 2011


STATUS

approved



