Where do the first two numbers of Fibonacci Sequence come from?

Question

I'm trying to code a simple algorithm that prints out the $n^{th}$ Fibonacci number. However, my program requires the initial seed values $F_0 = 0$ and $F_1 = 1$, when I'm hopeful I can figure something out to not require initial seed values.

Where does $F_0$ and $F_1$ come from anyway? Recursively, $F_n = F_{n-1} + F_{n-2}$, for all $n > 1$, but is there a mathematical approach to infer $F_0$ and $F_1$? Basically, how would I approach the values $F_0$ and $F_1$ without knowing them beforehand?

edit; I wanted to add since there seems to be confusion: I'm only concerned with the below sequence.

$$0, 1, 1, 2, 3, 5, 8, ...$$

I'm curious to how the $0, 1$ is defined, because every other term can be derived recursively, but the first two cannot (or can they?).

Answer 1

The initial values $F_0 = 0$ and $F_1 = 1$ are part of the definition of the Fibonacci sequence. They can't be derived, because e.g. you could just as well pick any two numbers and apply the same recursion relation to get a different sequence. They're just the simplest numbers to start with, in a sense.

In general, any recursive definition which specifies a member of a sequence in terms of the previous $n$ members will require you to put in the first $n$ values by hand. They can be chosen arbitrarily and then you use the formula to produce further values. If you know that you want to produce a sequence with certain properties, then perhaps you can "derive" the $n$ seeds you need to come up with such a sequence (although there is not guaranteed to be any more efficient way to do so than guessing and checking), but without some constraint on the sequence you expect to end up with, nothing specifies the seeds.

Answer 2

There is an entire family of Fibonacci-like sequences. A cool result by Wythoff is that for integral valued Fibonacci sequences, if you "center" them properly, you can use them to make a 1 to 1 correspondence with the rationals. However, the classic Fibonacci sequence is defined that way.

http://mathworld.wolfram.com/WythoffArray.html

Answer 3

If you want to "derive" the initial conditions in some sense you could do so in the idealized biological sense of rabbit pair populations like "in the book Liber Abaci (1202) by Leonardo of Pisa, known as Fibonacci."

http://en.wikipedia.org/wiki/Fibonacci_number#Origins

Answer 4

One compelling reason for the standard definition of the fibonacci sequence is that it reveals their interesting divisibility properties, namely that they form a strong divisibility sequence, i.e.

$$\rm\ gcd(f_m,f_n)\ =\ f_{\:gcd(m,n)}$$

hence $\rm\ m\ |\ n\ \Rightarrow\ f_m\ |\ f_n\ $ etc. Analogous results hold for the more general class of Lucas-Lehmer sequences, which prove convenient when studying elementary number theory of quadratic fields, e.g. generalizations of Fermat's little theorem, Euler's $\phi$ totient function, etc. If one changed the indexing then many of these results would be greatly obfuscated.

That said, it should be emphasized that such definitions are merely conventions that prove useful in the context at hand. In other contexts - where such divisibility properties play no role - another indexing might prove more convenient.

Answer 5

Technically it shouldn't matter which nonzero numbers you start with. The cool thing about Fibonacci number is the limit should go to the golden ratio.

Consider the definition of Golden Ratio - Smaller is to bigger piece, as bigger is to the whole.

As Fibonacci numbers get bigger and bigger, they get to this ratio. Intuitively speaking the original numbers matter less and less, and successive a, b, a+b patterns will go to:

a is to b, as b is to a+b.

Which gets closer and closer to the golden ratio.