The Fibonacci sequence is the simplest way to have an exponential in a discrete system. Discrete meaning non-continuous, as in whole numbers, integers.
As others have pointed out, there are multiple ways of doing this, but starting with 0 and 1 gives the simplest solution.
This is why it is said to be everywhere in nature. When something wants to grow exponential-like then it is a good chance that evolution settles on Fibonacci. This is especially true for more primitive plants.
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