This is elementary, but I found it somewhat surprising. Define $$ a_n = \frac{1+ a_{n-2}}{\sqrt{1+a_{n-1}}} \;,$$ where $a_1$ and $a_2$ are constants. For example, here is a plot for $a_1=5$ and $a_2=3$:
It appears that $\lim_{n\to\infty} a_n \approx 1.61803$, which is close to the golden ratio $\phi$. The limit appears to be independent of the two initial values $a_1$ and $a_2$, which is the aspect I found surprising. Of course $\frac{1+\phi}{\sqrt{1+\phi}}=\phi$, so it is not so surprising that the limit is $\phi$.
Q. For which values of $a_1$ and $a_2$ is $\lim_{n\to\infty} a_n = \phi$?