Can This Summation Formula Generate the Fibonacci Sequence?

[Vince00]

Homework Statement

Proof that $$\sum a, m=0,$$ (a-m)! / m!(a-2m)! = the fibonacci sequence.

Homework Equations

Fibonacci: 1, 1, 2, 3, 5, ... (but I think everyone knows that one!)

The Attempt at a Solution

Let Xm = $$\suma m=0$$ (a-m)! / m!(a-2m)!
I think I better proof Xm+2 = Xm+1 + Xm (follows from the fibonacci), so I can conclude that Xm = Fm+1.
I think using induction is the best way to go: Prooving it for X0 and X1, maybe even X2.
My attempt:
$$\sum a+2, m=0,$$ (a+2-m)! / m!(a+2-2m)! = $$\sum a+1, m=0,$$ (a+1-m)! / m!(a+1-2m)! + $$\sum a, m=0,$$ (a-m)! / m!(a-2m)!
=$$\sum a+1, m=0,$$ (a+1-m)! / m!(a+1-2m)! + $$\sum a+1, m=1,$$ (a-m-1)! / (m-1)!(a-m-1-m+1)! ... but I think I am making big mistakes here, because whatever I do, I get stuck.
I just don't see it!

Vince, freshmen physics, sorry but I don't know how to use the symbols and stuff!

 

[n!kofeyn]

It's difficult to read what you have written. Try editing it by using the proper LaTeX. Just click on my output below to see the way to do it.
$$\sum_{n=0}^\infty a_n$$ for sums and subscripts
$$\frac{(a-m)!}{m!(a-2m)!}$$ for fractions
Also, what is Xm?