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

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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.

$\blacksquare$


Sources