Among the first three terms, $1$ and $144$ are obviously square, but $14$ is not.

We claim that none of the rest terms will be a perfect square. This can be verified by using the MOD $16$ rule (see # 4144). All of the remaining terms are congruent to $12$ modulo $16$. Therefore, they cannot be squares. (Note that the remainder when being divided by $16$ is determined by the last three digits.)

In conclusion, the answer to this question is $\boxed{2}$.