Bounds on the number of squares in recurrence sequences: arbitrary $b$, III

We generalise our earlier work on the number of squares in binary recurrence sequences, ${\left\{y_k\right\}}_{k\ge -\mathrm{\infty }}$. In the notation of our previous papers, here we consider the case when $N_{\alpha }$ is any negative integer and $y_0=b^2$ for any positive integer, $b$. We show that there are at most $4$ distinct squares with $y_k$ sufficiently large. This allows us to also show that there are at most $9$ distinct squares in such sequences when $b=1,2$ or $3$, or once $d$ is sufficiently large.