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

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

That is:
 * $\sequence {a_n} = 2, 5, 13, 35, \ldots$

Then the generating function for $\sequence {a_n}$ is given as:
 * $\map G z = \dfrac 1 {1 - 2 z} + \dfrac 1 {1 - 3 z}$

Proof
Let $\map {G_2} z$ be the generating function for $\sequence {2^n}$.

Let $\map {G_3} z$ be the generating function for $\sequence {3^n}$.

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


 * $\map {G_3} z = \dfrac 1 {1 - 3 z}$

From Linear Combination of Generating Functions:
 * $\map G z = \map {G_2} z + \map {G_3} z$

Hence the result.