@r9m So, are the possible values for $y$ the following??
$$y^2=1, 3^2, 2^2, 2^2 \cdot 3^2, 2^4, 2^4 \cdot 3^2, 2^6, 2^6 \cdot 3^2$$
$$ \Rightarrow y=\pm 1 , \pm 3, \pm 2, \pm 2 \cdot 3, \pm 2^2 , \pm 2^2 \cdot 3 , \pm 2^3, \pm 2^3 \cdot 3 \\ \Rightarrow y=\pm 1 , \pm 3, \pm 2, \pm 6, \pm 4 , \pm 12 , \pm 8, \pm 24 $$
First thing we should note is:
$$\lim_{n\to\infty}\frac{U_n(x)^2}{U_{n-1}(x)^2+U_n(x)^2}=\lim_{n\to\infty}\frac{1}{\left(\dfrac{U_{n-1}}{U_n}\right)^2+1}$$
So, we look at $\displaystyle\lim_{n\to\infty}\frac{U_{n-1}}{U_n}$. Wikipedia gives the formula:
$$U_n=\frac{\left(x+\sqrt{x^2-1}\right)^{n...
