%I
%S 4,4,4,4,5,5,6,6,7,8,9,10,11,13,14,16,19,22,26,30,36,43,52,63,77,95,
%T 119,151,193,249,326,433,583,795,1102,1551,2220,3233,4796,7254,11194,
%U 17643,28432,46898,79271,137464,244869,448658,846699,1648170,3314300,6895838
%N Arises in enumerating iterated pointline configurations.
%C Lower bound of formula (12) on p.13. For some constants C1 and C2 the paper proves that C1*a(n) <= the number of points in the nth stage <= C2*(4^4^n). Abstract: Begin with a set of four points in the real plane in general position. Add to this collection the intersection of all lines through pairs of these points. Iterate. Ismailescu and Radoivi'c (2003) showed that the limiting set is dense in the plane. We give doubly exponential upper and lower bounds on the number of points at each stage. The proof employs a variant of the SzemerediTrotter Theorem and an analysis of the "minimum degree" of the growing configuration.
%H Joshua Cooper and Mark Walters, <a href="http://arxiv.org/abs/0807.1549">Iterated PointLine Configurations Grow DoublyExponentially</a>
%F a(n) = Floor(4^(1.0488^n)).
%e a(50) = 3314300 because 4^(1.0488^50) = 3314300.96.
%e a(51) = 6895838 because 4^(1.0488^51) = 6895838.31.
%e a(52) = 14869970 because 4^(1.0488^52) = 14869970.9.
%K easy,nonn,less
%O 1,1
%A _Jonathan Vos Post_, Jul 12 2008
