OFFSET
0,3
COMMENTS
Jul 02 2012: Duane DeTemple points out that one could argue that a(1) should be 0, not 1, since if the single point is removed from the plane, the result is not simply connected (and then the formula given below applies for all n). However, the sequence as described by Comtet only specifies "connected", not "simply connected", so I prefer to have a(1)=1. - N. J. A. Sloane, Jul 03 2012
n points in general position determine "n choose 2" lines, so a(n) <= A000124(n(n-1)/2). If n > 3, the lines are not in general position and so a(n) < A000124(n(n-1)/2). - Jonathan Sondow, Dec 01 2015
REFERENCES
L. Comtet, Advanced Combinatorics, Reidel, 1974, Problem 1, p. 72; and Problem 8, p. 74.
LINKS
T. D. Noe, Table of n, a(n) for n = 0..1000
Author?, Title? (from Alexander Evnin, Dec 06 2008)
Michal Opler, Pavel Valtr, and Tung Anh Vu, On the Arrangement of Hyperplanes Determined by n Points, EuroCG (39th European Workshop on Computational Geometry, Barcelona, Spain 2023) Session 7B, Talk 1, Vol. 54, No. 6.
Index entries for linear recurrences with constant coefficients, signature (5, -10, 10, -5, 1).
FORMULA
a(n) = (1/8)*(n-1)*(n^3-5*n^2+18*n-8) for n>1.
For n>1: a(0)=2, a(1)=7, a(2)=18, a(3)=41, a(4)=85, a(n)=5a(n-1)- 10a(n-2)+ 10a(n-3)-5a(n-4)+a(n-5). [Harvey P. Dale, May 06 2011]
For n>1, G.f.: (-2+3x-3x^2-x^3)/(-1+x)^5. [Harvey P. Dale, May 06 2011]
EXAMPLE
For n=2: draw three vertices forming a triangle and the three infinite straight lines joining them. There are a(3) = 7 connected regions.
MAPLE
A055503 := n->(1/8)*(n^4-6*n^3+23*n^2-26*n+8); [for n >1]
MATHEMATICA
Join[{1, 1}, Table[(1/8)(n-1)(n^3-5n^2+18n-8), {n, 2, 80}]] (* Harvey P. Dale, May 06 2011 *)
CROSSREFS
KEYWORD
nonn,nice
AUTHOR
N. J. A. Sloane, Jul 10 2000; Jul 03 2012
EXTENSIONS
a(1) changed from 0 to 1 by N. J. A. Sloane, Dec 07 2008
STATUS
approved