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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)product, which is sort of expected), but my technique is rusty at best.

So, the questions are: 1.Is it a known fact? If so, where can I read more on it? If not, may anybody help me to prove/disprove it?

PS:

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?

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# "Harmonacci" recurrence and identities for $\pi$

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), but my technique is rusty at best.

So, the questions are: 1. Is it a known fact? If so, where can I read more on it? If not, may anybody help me to prove/disprove it?

PS: 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. Does it mean anything?