TITLE: Where do the first two numbers of Fibonacci Sequence come from?
QUESTION [5 upvotes]: 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?).

REPLY [2 votes]: 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.