<s>[INST] Another question about the golden ratio and other numbers 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.,
$$\begin{eqnarray} L(8/5) &=&-1.69562076\ldots \newline L(13/8) &=& -1.76404686\ldots \newline L(21/13) &=& -1.68892398\ldots \newline L(34/21) &=& -1.68880982\ldots \end{eqnarray}$$
 [/INST] Some observations, but not a solution yet.
Let $t_k=\frac{F_{k+1}}{F_k}$ where $F_k$ is the $k^{th}$-Fibonacci number. The convention here for $F_k$ is that $F_3=2, F_4=3, F_5=5, \dots$. Denote the RHS in the above series by
$$\Psi_k(x)=\left(\sum_{n=1}^{\infty}\lfloor nt_k\rfloor x^{n-1}\right)^{-1}.$$
Then, it seems that
$$\Psi_k(x)=\frac{(1-x)(1-x^{F_k})}{P_k(x)}$$
for some polynomial $P_k(x)$ with the following rather curious properties:
(1) it has degree $F_k-1$;
(2) its coefficients are either $1$ or $2$ (although not clear which is which);
(3) $P_k(1)=F_k$;
(4) $L(t_k)=$ the smallest real root of $P_k(x)$;
(5) there might be a way to relate $(1-x)P_k(x)$ to other $(1-x)P_{\ell}(x)$ for $\ell<k$, recursively.
A few examples:
\begin{align}
\Psi_4(x)&=\frac{(1-x)(1-x^3)}{1+2x+2x^3} \\
\Psi_5(x)&=\frac{(1-x)(1-x^5)}{1+2x+x^2+2x^3+2x^4} \\
\Psi_6(x)&=\frac{(1-x)(1-x^8)}{1+2x+x^2+2x^3+2x^4+x^5+2x^6+2x^7} \\
\Psi_7(x)&=\frac{(1-x)(1-x^{12})}{1+2x+x^2+2x^3+2x^4+x^5+2x^6+x^7+2x^8+2x^9+x^{10}+2x^{11}+2x^{12}}.
\end{align}
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