by FEREGO
This web page is to contain various references pertaining to the Fibonacci numbers. There are also references to related results.
Fibonacci numbers appear not infrequently in mathematics and computer science. For example, there is a search called the “Fibonacci search.”
The Fibonacci numbers are those numbers that start with a 0 and a 1, and then continue by forming the sum of the previous two numbers. Thus we obtain the so-called Fibonacci sequence as 0, 1, 1, 2, 3, 5, 8, 13, and so on. As the references below clearly indicate, this seemingly simple sequence of numbers has some very remarkable properties.
Here are some Fibonacci number and related references:
Rabbit Story. Around the time frame of 1998 we wrote this fiction article about forest creatures.
Does this have anything to do with Tom Clancy's novel Red Rabbit? Some of this rabbit talk is frightening.
This article first appeared around the time frame of 1998, and we are reproducing it again in somewhat shortened (and somewhat simplified) form. The main body of the article though is essentially unchanged from what first appeared in 1998.
Other notes. The numbers referred to here have some interesting relationships to the Lucas numbers. The Lucas numbers are the numbers that begin with 1, 3, 4, 7, 11, etc., with the next number being determined as the sum of the previous two elements in the sequence. It is slightly better to begin this sequence with a 2, to form the sequence 2, 1, 3, 4, 7, 11, etc., so that some nice analytic properties are retained. The Fibonacci numbers are slightly different, and begin with a 1 and a 2 (or with a 0 and 1) to form the sequence 0, 1, 2, 3, 5, 8, 13, etc.
An interesting property of powers of the golden ratio (also known as the divine proportion [this was during the Middle Ages]) is that these powers become very close to the numbers in the Lucas sequence. (See the next item in this list.)
Pentium code. We have written computer codes to compute the Fibonacci (and generalized Fibonacci) numbers. We have developed a Pentium code (exe) that performs the additions using arbitrary precision integer arithmetic, and when this is run even on a slow Pentium these numbers get big very fast.
Mathematical derivations. We at FEREGO (and others) have derived some mathematical results pertaining to powers of the golden ratio. We have shown the relationship to the Lucas sequence, and have generalized the results. This work was originally done around 1990 in collaboration with someone else. Recently this work has been rewritten with some additions and improvements.
A really comprehensive compendium of Fibonacci facts is given at this reference. There are very many other references given on this site. This is a really good site pertaining to these interesting numbers. As you will see, there are a very large number of known results pertaining to the famous Fibonacci sequence.
I didn't know that that Fibonacci fellow was that leaning tower of Pisa guy. “Yes, I've heard of the leaning tower.”
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