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

1,2

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*Rho-1)^2 + 16*Rho*(k-1)) - 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*(R-2)+R/2+1); print1(ceil(N(k-1)), ", ")) \\ Hugo Pfoertner, Aug 02 2019

Crossrefs

Cf. A023393 (best known packings).

Sequence in context: A331952 A225386 A037258 * A024521 A194455 A163324

Adjacent sequences: A201990 A201991 A201992 * A201994 A201995 A201996

Keyword

nonn

Author

Hugo Pfoertner, Dec 07 2011

Status

approved