Here from the fraction set we have a really hard question to be answered...suppose that a sequence is defined as $a_{n} = a_{n-1} - \dfrac 1{a_{n-1}}$, where $a_0$ is given. ...you already know what I'm asking you ... find the $n^{\text{th}}$ element of the sequence in terms of $n$ and $a_0$...fun! :)
P.S I've been trying for a lot of time ... but until now I haven't managed to find a solution; only some very weird graphs which can puzzle you...if you wish I can send you some of them!
It's very unlikely that a closed-form solution can be found. $a(n)$ is a rational function of $a(0)$ with numerator of degree $2^n$ and denominator of degree $2^n-1$.