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From: davids@utcsstat.UUCP (David Scollnik)
Newsgroups: net.math
Subject: Pascal's Inverse Triangle
Message-ID: <2216@utcsstat.UUCP>
Date: Wed, 3-Jul-85 12:55:32 EDT
Article-I.D.: utcsstat.2216
Posted: Wed Jul  3 12:55:32 1985
Date-Received: Wed, 3-Jul-85 13:52:43 EDT
Organization: U. of Toronto, Canada
Lines: 70



Undoubtedly, the great majority of you are familiar with Pascal's
             triangle, that is, this triangle which continues off
             to infinity ...

                                 1
                               1   1
                             1   2   1
                           1   3   3   1
                         1   4   6   4   1
                       1   5   10  10  5   1
                     1   6   15  20  15  6   1

              note, every entry in the triangle is the sum of
                    the two entries directly above.

Now, consider this, which I shall refer to as Pascal's Inverse triangle...


                                  1
                               1     1
                             1   1/2   1
                           1   2/3  2/3  1
                         1   3/5  3/4  3/5  1

                this triangle also continues on for infinity.
           
      Each entry is the INVERSE of the sum of the two entries 
       directly above, for example

                                     1/2 = 1/(1 + 1)

                              and    2/3 = 1/(1/2 + 1)
 
                              and    3/4 = 1/(2/3 + 2/3) .

     Of course, the outermost layer will always be one,
                and it can be shown (for example with the
                use of continued fractions) that the next 
                layer inward will converge to (sqrt(5)-1)/2.

     This is apparent since we have, as the number of levels in
        the triangle goes off to infinity, that


                                        1
                            X   =  -----------------
                                    1   +   1
                                          -----------
                                            1  +  1
                                                 ---------
                                                   1  +  etc.

                and this is the same as
          

                                        1
                           X  =  ---------------
                                    1   +   X

                                X * X  +  X  -  1 = 0

                        and this equations one positive root is

                                 (sqrt(5)-1)/2

           Has anyone come across this triangle before, and if so
               have you anything of interest to share concerning it ?