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About $a_n = \frac{a_{n-1}(a_{n-1} + C)}{a_{n-2}}.$
Let $A,B,C > 0$. Put $a_1 = A$ and $a_2 = B$ and, for integer $n > 2$,
$$a_n = \frac{a_{n-1}(a_{n-1} + C)}{a_{n-2}}$$
and
$$ T = \lim_{k \to \infty} \frac{a_k}{ a_{k - 1}}.$$
Notice the limit seems …