Explicit a succession using z inverse transform
If you are allowed to define your system to be causal so that it is non-zero only for $n \ge 0$, then you can use the unit step function $u[n]$ so that your system becomes:

$$y[0] = 1$$
$$y[n+1] = 2y[n] + 3u[n]$$

Which does have a Z-transform representation.