From: Per Erik Manne <per@hamilton.nhh.no>
Newsgroups: sci.math
Subject: Re: Find a formula for the series : 1, 2, 4, 8, 16, 31
Date: Tue, 12 May 1998 14:46:30 +0200

H. M. Shannon wrote:
> 
> In article <354DB150.41C6@hamilton.nhh.no>,
> Per Erik Manne  <per@hamilton.nhh.no> wrote:
> >Ronald Bruck wrote:
> >>
> >> Most likely, the original poster wants:
> >>
> >>   f(n) = sum of binomial(n,p) from p = 0 through 4.
> >>
> >> This is similar to the problem of the number of regions a circle is cut
> >> into when n points on its circumference are joined by lines.  (But it's
> >> not exactly that, since it's got one more term.)
> 
> There's a rather neat argument to this effect in "The Book of Numbers"
> (Conway & Guy; I'm sure you already know it), p. 76-79.  Have fun.
> 
> (Try n-1 choose p, by the way.)
> 
> >I've heard f(n) described as the maximal number of pieces you can cut
> >a cake into with n straight cuts when the cake is 4-dimensional.
> 
> I'd be interested in hearing a generalization to higher dimensions...
> 
> --Heather M. Shannon

Coxeter: Introduction to Geometry (second edition, p.183) has
the following quote attributed to Ludwig Schlafli (1814-1895):

  "If i hyperplanes in n dimensions are so placed that
   every n but no n+1 have a common point, the number of
   regions into which they decompose the space is

     (i over 0) + (i over 1) + (i over 2) + ... + (i over n)

     = f(n,i)."

The proof of this assertion is developed in the exercises
in Coxeter's book (p.186) with a reference to Jacob Steiner
(1796-1863) crediting him with the case of 3-dimensional space.
--
Bergen,
Per Manne
--
Bergen,