Fibonacci sequence: how does $0$ get to $1$?

Question

In the Fibonacci sequence, how does $0$ get to $1$?

$$ 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, \ldots$$

The rule is adding the previous $2$ numbers, and the previous $2$ numbers before $1$ are $0$ and $-1$.

$$0 + (-1) = -1.$$

So how does it get to 0 - 1? Just interested.

Answer 1

Nope, the previous 2 numbers before 1 are 0 and 1. If we prolong your sequence...

$$\ldots, -8, 5, -3, 2, -1, 1, 0, 1, 1, 2, 3, 5, 8, \ldots$$

Notice a pattern? :)

Answer 2

The $n$-th Fibonacci number is given by the equation $F_n = F_{n - 1} + F_{n - 2}$, which means, in effect, "add the last two numbers of the sequence". But if someone asks you for the first term of the sequence, you cant tell them to "add the 0-th and -1-st numbers of the sequence", because those aren't defined. The same problem arises if someone asks for the second term of the sequence. So we have to define the first and second terms of the sequence ourselves. The most obvious choice to make is $F_1 = 0$ and $F_2 = 1$. Those are called "seed values", and nothing says they have to be 0 and 1. In fact, interesting things happen when you choose different seed values.

Answer 3

One usually considers the Fibonacci sequence to start at $F_0$, so that there is no pair of values preceeding $F_1$ (nor is there for $F_0$). This means that $F_0$ and $F_1$ must be given explicitly as part of the definition, in addition to the recurence relation which applies only for $n\geq2$. In fact they are given by $F_0=0$ and $F_1=1$; fixing them in another way would give a Fibonacci-like sequence such as for instance the Lucas numbers, but not the Fibonacci sequence itself.

As is indicated in other comments and answers, you can extend the Fibonacci sequence to negative indices (which one might still call the Fibonacci sequence as no existing values are changed, but it really becomes a different sequence, doubly infinite this time). But then the values at negative indices are computed using the recurrence relation backwards, rewriting it as $F_{n-2}=F_n-F_{n-1}$. But this (still) does not explain the values of $F_0,F_1$ from its predecessors, but to the contrary explains those predecessors in terms of $F_0,F_1$.