The golden ratio solves the equation:
$$x^2-x-1=0$$
Equivalently: $x-(1/x)=1. $
What about generating a sequence of ratios $R_n$, by the equation: $x-(1/x)=n $
Has this been studied?
We would have $R_0=1$, and depending on taking large/small, the sequences:
$1, 1.618..., $
$1, 0.618..., $