

A140744


Arises in enumerating iterated pointline configurations.


1



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, 119, 151, 193, 249, 326, 433, 583, 795, 1102, 1551, 2220, 3233, 4796, 7254, 11194, 17643, 28432, 46898, 79271, 137464, 244869, 448658, 846699, 1648170, 3314300, 6895838
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OFFSET

1,1


COMMENTS

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.


LINKS

Table of n, a(n) for n=1..52.
Joshua Cooper and Mark Walters, Iterated PointLine Configurations Grow DoublyExponentially


FORMULA

a(n) = Floor(4^(1.0488^n)).


EXAMPLE

a(50) = 3314300 because 4^(1.0488^50) = 3314300.96.
a(51) = 6895838 because 4^(1.0488^51) = 6895838.31.
a(52) = 14869970 because 4^(1.0488^52) = 14869970.9.


CROSSREFS

Sequence in context: A006264 A134994 A138195 * A179414 A139324 A111655
Adjacent sequences: A140741 A140742 A140743 * A140745 A140746 A140747


KEYWORD

easy,nonn,less


AUTHOR

Jonathan Vos Post, Jul 12 2008


STATUS

approved



