3n^2-1 =  x^2 is impossible for all integers x.

3n^2-1 = x^2 => 3 divides x^2 +1 => x = 3m+1 or x = 3m+2.

Case x=3m+1.

 3n^2  = 9m^2+6m+2 = 3r+2 a contradiction. So, the statement is true for all squares of the form 3n+1.  

Similarly, for x = 3m+2 we have 3n^2 = 9m^2+12m+5 = 3r+5 = 3s+2 another contridiction. So the statement is true for all squares of the form 3m+2.

Cino Hilliard