This is related to Pisano periods, that is, the periods of the Fibonacci numbers modulo $k=2, 3, \cdots$. I am studying the sequence $x(n+1)=\{b x(n)\}$ (here the brackets represent the fractional part function) with $b=(1+\sqrt{5})/2$ and $x(1) = 1/k, k=2, 3, \cdots$. These sequences are also periodic.
I tried several values of $k$ and in all the cases both periods (Pisano periods and periods from my sequences) were identical. Is this a coincidence, or a well known result easy to prove? For the context, read my article on randomness theory, here. This fact is discussed in section 3.3.(b). My sequence is associated with the golden ratio numeration system, a well known numeration system with an irrational base.