Consider the sequence $\displaystyle (a_n)_{n\ge 0}$ defined by the following recurrence

$\displaystyle a_0 = 2, a_{n+1} = 4a_n + 5, \forall n \ge 0.$

Let $\displaystyle f(z) = \displaystyle\sum_{n=0}^{\infty}a_nz^n = a_0+a_1z+a_2+z^2+.... $ be the gernerating function for the sequence.

Express f(z) as a rational function.