Exploration for Fibonacci Numbers

    In Mathematica, the Fibonacci numbers can be defined recursively as follows:

Caveat.  When using a recursion, a function call usually computes the value making all of the necessary previous recursive steps.  Sometimes it is faster to store all the previous values so that a second function call can use information previously calculated.  If the Mathematica function is written in the following way, it includes this "remember" feature.  

Without the "remember" feature the function is:

Global`F
 

Let us compute the Fibonacci number  

The function can be viewed with the command:

Global`F
 

Now let us add the "remember" feature to the function:

Let us compute the Fibonacci number  

The function can be viewed with the command:

Global`F
 

We see that all of the Fibonacci numbers   through are actually saved in Mathematica and if a second call to them is made then the value will be retrieved from memory.

Another nice way to define the Fibonacci numbers is using a function defined on subscripts.

Closed form formula for Fibonacci numbers

Using the standard technique for solving a difference equation, it is easy to show that can be computed with the formula

        

Limit of the ratio of  Fibonacci numbers

The ratio of consecutive Fibonacci numbers is

        

The limit of this ratio is    as shown by the computation  

But this is just a disguised way of writing the value    for the "golden ratio."

Since the Fibonacci search uses the ratios   when  n  large the two methods are almost identical.  

Even for the small value    the first six values of    are very close to the golden ratio

(c) John H. Mathews 2004