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From: wjhe@hlexa.UUCP (Bill Hery)
Newsgroups: net.math
Subject: Re: Pascal's Inverse Triangle
Message-ID: <4375@hlexa.UUCP>
Date: Mon, 8-Jul-85 14:56:25 EDT
Article-I.D.: hlexa.4375
Posted: Mon Jul  8 14:56:25 1985
Date-Received: Tue, 9-Jul-85 06:38:23 EDT
References: <2216@utcsstat.UUCP>
Organization: AT&T Bell Laboratories, Short Hills, NJ
Lines: 48

> 
> 
> Undoubtedly, the great majority of you are familiar with Pascal's
>              triangle, that is, this triangle which continues off
>              to infinity ...
> 
>...... 
> 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
> 
>            
>       Each entry is the INVERSE of the sum of the two entries 
>        directly above, for example
>........ 
>      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.
>......... 
>            Has anyone come across this triangle before, and if so
>                have you anything of interest to share concerning it ?

The terms in the 'next layer' (if the fractions are not reduced ) 
can be shown to be the inverse ratio of the successive terms of the Fibonacci
series (a(0)=1, a(1)=1, a(n)=a(n-1)+a(n-2) for n>1); the limit of this ratio
is well known (and easily shown to be) (sqrt(5)-1)/2.

The Fibonacci series (actually a sequence in modern mathematical parlance)
comes up in many places in nature (e. g., the number of petals in succesive
layers in a flower), and has been studied by mathematicians for many hundreds 
of years.  There is a society (called the Fibonacci Society) which is devoted 
to it's study, and publishes a scholarly journal (the Fibonacci Quarterly)
which publishes articles on the theory and applications of Fibonacci Series.
I suggest consulting that journal for possible leads.

Another source would be texts on finite difference equations, of which
the general form for an element of your triangle is an example.  This area
has been somewhat dormant for a while, but Dave Jaegerman of ATT Bell Labs
(Holmdel NJ) is about to release a new book on the topic.

Bill Hery
ATT Bell Labs