Is there a way to determine a formula giving all *integer* values of $x$ for which the value of a polynomial $P(x)$ with *integer coefficients* is a square?

That is, is there a closed formula for:

$X = \{ x \in \mathbb{N} : \exists \ n \in \mathbb{N} : P(x) = n^2 \}$ ?

I'm interested in particular in $P'(x) = 8x^2-8x+1$, but am wondering about the general case as well.

For $P'(x)$ a sample of $X$ is $\{ 1, 3, 15, 85, 493, 2871, 16731, 97513, \ldots \}$.