I am having a hard time generating a characteristic polynomial for a recurrence relation I thought of the other day, $a_n = a_{n-1} + \frac1{a_{n-1}}$.
I am pretty familiar solving basic recurrence relations such as $F_n = F_{n-2} + F_{n-1}$ being $x^2 = x + 1$, but this solution seems to escape me.
I recognize the fact that this will probably look something akin to a logarithm because we are constantly adding smaller and smaller $1/x$, but beyond that I am lost.
I do not know how to approach this, and my discrete math is a bit rusty.. Any help would be appreciated.