Generating Function for Sequence of Powers of Constant/Examples/2^n + 3^n

Example of Generating Function for Sequence of Powers of Constant
Let $\left \langle {a_n}\right \rangle$ be the sequence defined as:
 * $\forall n \in \Z_{\ge 0}: a_n = 2^n + 3^n$

That is:
 * $\left \langle {a_n}\right \rangle = 2, 5, 13, 35, \ldots$

Then the generating function for $\left \langle {a_n}\right \rangle$ is given as:
 * $G \left({z}\right) = \dfrac 1 {1 - 2 z} + \dfrac 1 {1 - 3 z}$

Proof
Let $G_2 \left({z}\right)$ be the generating function for $\left \langle {2^n}\right \rangle$.

Let $G_3 \left({z}\right)$ be the generating function for $\left \langle {3^n}\right \rangle$.

From Generating Function for Sequence of Powers of Constant:
 * $G_2 \left({z}\right) = \dfrac 1 {1 - 2 z}$


 * $G_3 \left({z}\right) = \dfrac 1 {1 - 3 z}$

From Linear Combination of Generating Functions:
 * $G \left({z}\right) = G_2 \left({z}\right) + G_3 \left({z}\right)$

Hence the result.