How to find inverse z transform
Using binomial formulas one easily gets
$$
Y(z)=\frac{\frac12+z^{-1}}{1-\frac14z^{-2}}
=(\tfrac12+z^{-1})\sum_{k=0}^\infty 2^{-2k}z^{-2k}.
$$
Now compare to the coefficients in $Y(z)=\sum_{n=0}^\infty y_nz^{-n}$ for $n=2k$ and $n=2k+1$, even and odd indices.