When is $32x+32$ a square number? I am trying to find out for what values for $x$ does the function $f(x)=32x+32$ return a square number? I found that this is the case for at least: $x \in \{64,1600,53824,1827904,62094400,2109381184\}$ but I can't see a pattern in those numbers.

Is there a formula which generates these values for $x$ instead of simply trying a lot of values?
$$32(x + 1) = y^2 \implies 32 \mid y^2 \implies 4 \mid y$$ so let $y = 4z$. $$2(x+1) = z^2 \implies 2 \mid z$$ so let $z = 2w$ $$x + 1 = 2w^2 $$ so given any $w$ we can solve for $x$.

So the set of solutions is
$$\{(2w^2 - 1, 8w) \mid w \in \Bbb Z\}$$
Solution for x such that 32x+32 is a square as follows:
 For integral solution,  x= (n2 -2)/2 such that x is in Natural number.(n is in natural number)