From: radcliff@alpha2.csd.uwm.edu (David G Radcliffe)
Newsgroups: sci.math
Subject: Re: In the Region of a Pointless Question?
Date: 11 Feb 1997 05:44:27 GMT

Mike Bernstein (mbernstein@clara.net) wrote:
: Please DOES ANYONE KNOW if there is a relationship between the number of
: equidistant points on the circumference of a circle and the number of
: regions generated by joining, with straight lines, every point to every
: other point.
: 
: Mike Bernstein


It is easy to count the number of regions when n is small.

# of points:   1   2   3   4   5   6
# of regions:  1   2   4   8  16  30

Has anybody studied this sequence before?  A good place to check is the 
Encyclopedia of Integer Sequences.  I sent an email message to 
sequences@research.att.com with the line "lookup 1 2 4 8 16 30" in the 
message body (no quotes).  A few minutes later, I received the following 
response:
============================================================================
%I A006533 M1118
%S A006533 1,2,4,8,16,30,57,88,163,230,386,456,794,966,1471,1712,2517,2484,
%T A006533 4048,4520,6196,6842,9109,9048,12951,14014,17902,19208,24158,
%U A006533 21510,31931,33888
%N A006533 Join $n$ equal points around circle in all ways, count regions.
%R A006533 WP 10 62 72. PoRu94.
%O A006533 1,2
%Y A006533 Cf. A007678.
%A A006533 njas, Bjorn Poonen (poonen@math.princeton.edu)
%K A006533 nonn,easy

References (if any):

[PoRu94] = B. Poonen and M. Rubinstein, ``The number of intersection points made
 by the diagonals of a regular polygon,'' preprint, 1994.    
============================================================================

[Notice that the sequence is not monotone!  Intriguing...]

A web search (using AltaVista) immediately locates this preprint. 
The URL is http://www.msri.org/MSRI-preprints/online/1995-060.html .
The article does give an explicit formula, but it is very complicated,
and I do not have time to untangle it. 

-- 
David Radcliffe				radcliff@alpha2.csd.uwm.edu