Michael Thomason's write-up for assignment twelve, question four.

Generate a Fibonacci sequence in the first column using f(0) = 0, f(1) = 1, f(n) = f(n-1) + f(n-2).

The first two numbers are called seeds. Using different seeds changes the sequence, but we will see that the ratio between each iteration (step) and the one before it tends to the same number regardless of which two seeds are used.

Here I have created an Excel worksheet that calculates the first several iterations for the indicated seeds. You may open it by clicking here; change the seeds and press enter to see what happens to the sequence.

The first Fibonacci sequence in the above table is quite well-known and is shown here:

Do the above numbers increase by a proportion determined by the seeds, or does their progression continue by the same proportion regardless of the seeds chosen? Here is another Excel spreadsheet, this time with the ratio between each step displayed:

(Click here to change the seeds and observe how  behaves.)

For all three sequences, the value for  tends to the same number, 1.61803399. This number is called the golden ratio, and is exactly expressed as . Click here to see the first 5000 digits of the golden ratio, which is also called  (phi).

This sequence, for which the seeds are f(0)=1 and f(1) = 3, is called a Lucas sequence. It too has values for  tending to .