While playing with something totally irrelevant I stumbled upon the recurrence: $$a_{n+1} = \frac{1}{a_n} + a_{n-1}$$

It turns out that given $a_0 = 1, a_1 = 1$,

$$lim \frac{a_{2n}}{a_{2n-1}} = \frac{\pi}{2}$$

I have a very crude idea (or rather a hint) on proving it (the iterations sort of unfold into a sort of Viete product, which is sort of expected), but my technique is rusty at best.

With different initial conditions, things start getting really scary, for example $ a_0 = 2, 3, 4, 5 $ yield $\frac{8}{\pi}, \frac{9\pi}{8}, \frac{128}{9\pi}, \frac{225\pi}{128}$ respectively.

So, the questions are: Is it a known fact? If so, where can I read more on it? If not, may anybody help me to prove/disprove it? Does it mean anything?