Inverse $\mathcal Z$-transform when region of convergence goes outwards from the inner pole?
There exist $3$ sequences with the given expression as their $\mathcal{Z}$-transform. These $3$ sequences correspond to $3$ different regions of convergence (ROCs):


$|z|&gt;\frac12$: right-sided
$\frac13&lt;|z|&lt;\frac12$: two-sided
$|z|&lt;\frac13$: left-sided


It's important that you figure out both right-sided and left-sided inverse $\mathcal{Z}$-transforms of the basic transform

$$X(z)=\frac{1}{1+az^{-1}}$$

and their respective ROCs. Then it's easy to find all $3$ inverse transforms for the given example.