Why the difference? And when we're deriving these series from eigenvectors, what difference does the starting point make?
Please help. I'm very confused. I have a test tomorrow and need to know the answer to this question. There's literally nothing in my book and I couldn't find anything on the internet either.
On
There is at least one nice property of the Fibonacci numbers that depend on the indexing. The Fibonacci numbers form a "divisibility sequence" - that is, if $m | n$ then $F_m | F_n$. (If $m = 3$, for example, this is the fact that if $n$ is divisible by 3 then $F_n$ is even.) This doesn't hold if the Fibonacci sequence is indexed differently.
A lot of the relationships between Lucas and Fibonacci numbers, in turn, seem to be most cleanly expressed if the Lucas numbers are indexed as they are, for example $L_n^2 = 5 F_n^2 + 4 (-1)^n$.
If you're concerned about deriving these from the eigenvalues, though, I don't think any of this matters.
The difference equation $$x_{n+2}-x_{n+1}-x_n=0\qquad(n\in{\mathbb Z})$$ has a two dimensional solution space ${\cal L}$ of functions $x:\>{\mathbb Z}\to{\mathbb R}$. The initial conditions $x_0=0$, $x_1=1$ determine a nontrivial solution $(F_k)_{k\in{\mathbb Z}}$ called the Fibonacchi sequence. In order to obtain a basis of ${\cal L}$ we need a second solution which is linearly independent of $(F_k)_{k\in{\mathbb Z}}$. Initial conditions of the form $x_0=0$, $x_1=c$ would just produce a constant multiple of $(F_k)_{k\in{\mathbb Z}}$. We therefore have to start with initial conditions enforcing $x_0\ne0$. It seems that for the Lucas sequence one chooses $x_0=2$, $x_1=1$.