This is an extension of Mathoverflow #258155, "A question about the golden ratio and other numbers."  Suppose that $$c_0+c_1x+c_2x^2+ \cdots = \frac{1} {\lfloor{r}\rfloor+\lfloor{2r}\rfloor  x+\lfloor{3r}\rfloor x^2+ \cdots},$$
where $r = (1+ \sqrt{5})/2$.  Can someone prove that the number 
$$L(r) = \lim \frac {c_{i+1}} {c_{i}}$$ exists?   It appears that $$L(r)  = -1.688924110769165206686359\ldots$$  Also, it appears that $L(F_{k+1}/F_{k})$ exists, for $k \ge 5$, where $F_k$ denotes the $k$th Fibonacci number; e.g.,
$$L(8/5) =-1.69562076\ldots$$
$$L(13/8) = -1.76404686\ldots$$
$$L(21/13) = -1.68892398\ldots$$
$$L(34/21) = -1.68880982\ldots$$