Generating function of the sequence: $a_n = \frac{(6^n+1)^2}{2^n}.$

Question

Find the generating function of the sequence with $$a_n = \frac{(6^n+1)^2}{2^n}.$$

First of all I writed it like that

$\displaystyle G(x) =\sum\limits_{n=0}^\infty\left(\frac{(6^n+1)^2}{2^n}\right)x^n$

Then, it is equal to

$\displaystyle G(x) =\sum\limits_{n=0}^\infty\left(18^n+2\cdot3^n+ \frac{1}{2^n}\right)x^n$

I don't know how can I continue after that

Answer 1

HINT: Split the summation into three summations, and use the fact that

$$\sum_{n\ge 0}a^nx^n=\sum_{n\ge 0}(ax)^n=\frac1{1-ax}\;.$$