Exercise 13.

You may assume that \(1, 1, 2, 3, 5, 8,\ldots\) has generating function \(\dfrac{1}{1-x-x^2}\) (because it does). Use this fact to find the sequence generated by each of the following generating functions.

1. \(\frac{x^2}{1-x-x^2}\text{.}\)

2. \(\frac{1}{1-x^2-x^4}\text{.}\)

3. \(\frac{1}{1-3x-9x^2}\text{.}\)

4. \(\frac{1}{(1-x-x^2)(1-x)}\text{.}\)