For Fibonacci sequence,

             x = 1x^1
      x * g(x) =        1x^2 + 1x^3 + 2x^4 + 3x^5 + 5x^6 + ...
  + x^2 * g(x) =               1x^3 + 1x^4 + 2x^5 + 3x^6 + ...
  ------------------------------------------------------------
  =       g(x) = 1x^1 + 1x^2 + 2x^3 + 3x^4 + 5x^5 + 8x^6 + ...

Hence,

     x = (1 - x - x^2) * g(x)
  g(x) = x / (1 - x - x^2)

Generating functions are amazing. One of the coolest topics in my entire undergrad math degree. This PDF is well written and will explain everything: http://courses.csail.mit.edu/6.042/fall05/ln11.pdf