Let $(u_n)$ be the sequence defined by $u_{n+2}=\frac{u_{n+1}}{u_n}+\frac{u_n^2}{u_{n+1}^2}$ for $n\geq 1$.
When $u_1=u_2=1$, by using a calculator, I noticed that the even subsequence converges to 2.2287 and the odd subsequence converges to 1.8997.
When $u_1=3$ and $u_2=2$, by using a calculator, I noticed that the even subsequence converges to 1.9479 and the odd subsequence converges to 2.9384.
In fact, no matter which positive real numbers that I set for $u_1$ and $u_2$, the even and odd subsequences of $(u_n)$ converges (to their respective limits). I tried, rather naively, by back substitution to find the closed form in an attempt to prove this conjecture but failed due to the expressions being too tedious.
Would appreciate any insight, like some ideas of proof if the conjecture is correct or a counter example of positive $u_1$ and $u_2$ that disproves the conjecture.