Consider the following recursions

$$ x_{n+2} = x_{n+1} + \frac{x_n}{n} $$

$$y_{n+2} = \frac{ y_{n+1}}{n} + y_n $$

I have been toying around with different starting values ( complex Numbers ) , divergeance etc. But was not able to conclude much.

However I noticed when

$$ x_1 = 0 $$

$$y_1 = 0 $$

$$ x_2 = 1 $$

$$ y_2 = 1 $$

We get the following limit recursions

$$ \lim_{n \to \infty} \frac{n}{x_n} = e $$

$$ \lim_{n \to \infty} \frac{2 n}{y_n ^2} = \pi $$

How to prove these ??

And how about the divergeance / convergeance for other complex initial values ?

Edit : a partial answer occurs here

Mirror algorithm for computing $\pi$ and $e$ - does it hint on some connection between them?

http://www.pi314.net/eng/miroir.php

But the issue of other starting values is not resolved yet.

( so this is not a complete duplicate )

For the first recursion we have an answer ( see below ) but at the time of posting , the second has no answer with respect to variable initial conditions yet.