This is an extension of "A question about the golden ratio and other numbers." Given $r$, suppose that $$c_0+c_1x+c_2x^2+ \cdots = \frac{1} {\lfloor{r}\rfloor+\lfloor{2r}\rfloor x+\lfloor{3r}\rfloor x^2+ \cdots}.$$ Let $L(r) = \lim_{i\to\infty} \frac{c_{i+1}}{c_i}$. Can someone prove that the limit $L(\phi)$ exists, where $\phi = \frac{1+ \sqrt{5}}{2}$? It appears that $$L(\phi) = -1.688924110769165206686359\ldots$$

$$(c_0,c_1,c_2,\ldots) = (1,−3,5,−9,17,−30,52,−90,154,−262,446,−758,1285,\ldots).$$

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., $$\begin{eqnarray} L(8/5) &=&-1.69562076\ldots \newline L(13/8) &=& -1.76404686\ldots \newline L(21/13) &=& -1.68892398\ldots \newline L(34/21) &=& -1.68880982\ldots \end{eqnarray}$$