A140744 Arises in enumerating iterated point-line configurations.

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

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 n-th 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 Szemeredi-Trotter 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 Point-Line Configurations Grow Doubly-Exponentially

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 A361248 A139324

Adjacent sequences: A140741 A140742 A140743 * A140745 A140746 A140747

Keyword

easy,nonn,less

Author

Jonathan Vos Post, Jul 12 2008

Status

approved