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From: lambert@boring.UUCP
Newsgroups: net.math
Subject: Re: Counter-Intuitive Sequences
Message-ID: <6761@boring.UUCP>
Date: Wed, 5-Feb-86 08:20:04 EST
Article-I.D.: boring.6761
Posted: Wed Feb  5 08:20:04 1986
Date-Received: Fri, 7-Feb-86 21:36:04 EST
References: <748@garfield.UUCP> <8675@ucla-cs.ARPA>
Reply-To: lambert@boring.UUCP (Lambert Meertens)
Organization: CWI, Amsterdam
Lines: 34
Apparently-To: rnews@mcvax

In article <8675@ucla-cs.ARPA> verma@ucla-cs.UUCP (Thomas S. Verma ) writes:
> In article <748@garfield.UUCP> robertj@garfield.UUCP (Robert Janes) writes:
>> Consider the following sequence: 1, 2, 4, 8, 16,.... what is the next term?	
>> Invariably the response is "32".
>> This however does not have to be the case and an alternate sequence arises
>> very naturally. Consider the sequence of n where n is the number of regions
>> the interior of a circle can be divided into using k lines where k starts
>> at 0. If you do that for the first five k you get the above sequence.
>> Suggestive isn't it ? However it turns out that for k=6 n=31 and the 
>> intuitive result falls flat on its face.
> 
> I do not understand what you mean.  I drew a circle.  I saw one region.
> I then drew a chord.  I saw two regions.  I drew another chord.  I saw
> four regions.  Now the third chord could go in two non-isomorphic places.
> One gave six regions, the other gave seven.  [...]
> Please tell me what you are talking about...

Undoubtedly, robertj was referring to the sequence mentioned in
article <1226@lll-crg.ARpA> by cralle@lll-crg.UUCP (R.(Bob) K. Cralle):

> [...] and so the series goes:

> 1 2 4 8 16 31 57 99 163 256 386 562...  [isn't the 256 curious--one late!]

> What does the series represent? If you have n points on a circle & and you
> connect every point to every other point the series gives the number of areas
> those lines cut the circle into. (more than two lines intersecting at a point
> not allowed.)

-- 

     Lambert Meertens
     ...!{seismo,okstate,garfield,decvax,philabs}!lambert@mcvax.UUCP
     CWI (Centre for Mathematics and Computer Science), Amsterdam