The legend, probably true, has it that Fibonacci came up with his numbers while observing how the population of rabbits grows due to their breeding. Even though this was centuries ago, I have this uneasy feeling that rabbits still don't get it. And I am quite sure that most traders don't get it either.
I have expressed my strong opinion, somewhat negative, about using Fibonacci indicators for trading in the not so distant past. This opinion, while expressed in the context of e-mini futures, my favorite trading instrument, applies to trading other financial instruments as well, and has more to do with how these indicators are often used (if not abused) and less with the indicators themselves.
Now, what I really want to talk about in this article is how to obtain Fibonacci numbers (1, 1, 2, 3, 5, 8, 13, 34, and so on) without rabbits. First, because it only confuses them, the rabbits, and second, because you really don't need two rabbits (and of opposite sex at that) to derive Fibonacci numbers.
You only need one, meaning number 1. Instead of two numbers (say 0 and 1 or 1 and 1) that would form the beginning of the Fibonacci sequence to be obtained via the recursive formula, a(n)=a(n-1)+a(n-2). The formula in question, if applied correctly does produce all the Fibonacci numbers.
For instance, a(2)=a(1)+a(0)=1+0=1, a(3)=a(2)+a(1)=1+1=2, a(4)=a(3)+a(2)=2+1=3, etc.
How can we do this then?
Well, take 1 and add the sum of all the previous numbers, starting from 1, to it.
This sum is 1 for the first term, so the next term we obtain this way is 1+1=2. We now have two terms, 1 and 2, and we can repeat the same procedure obtaining 2+(1+2)=5, as our sum is now 1+2. And since we now have three terms, 1, 2, 5, we can use them to obtain the next term: 5+(1+2+5)=13. In the next step, we have 13+(13+5+2+1)=34.
What we are getting is 1, 2, 5, 13, 34,...
Now, if you know the Fibonacci sequence, or saw a piece of it above, you may be feeling a bit uneasy because that really does not very much resemble one.
Well, only partially so. All these numbers are Fibonacci numbers, but it is also true that these numbers are not all Fibonacci numbers, not the complete sequence of them. It's only every other number from this famous sequence.
And that's fine, because if you really want to obtain the full Fibonacci sequence, you can do this quite easily using only the numbers you have obtained so far and all those that you can continue obtaining in the way outlined above.
You simply subtract the numbers obtained, so 2-1=1 and it goes between 1 and 2, then 5-2=3 and it goes between 2 and 5, and 13-5=8 that goes, obviously, between 5 and 13.
In other words, we can obtain the entire Fibonacci sequence starting just from a single number, 1. Neat, isn't it?
But the sequence of every other Fibonacci number is even easier to obtain (no subtraction needed) starting just from 1 and hence perhaps it is even more fundamental in some way.
Incidentally, I have not seen this kind of derivation before, although I am rather familiar with the Fibonacci literature, but perhaps someone else has come up with it before me, so I will abstain from claiming any priority. Not to mention that this is, at the very best, just a piece of recreational mathematics.