Approximation of harmonious numbers I have a question about an approximation of so-called "harmonious numbers". These are a generalisation of the golden ratio, the plastic number, and so on, (related to the Fibonacci and Padovan sequences respectively). Their decimal expansions to $5\ sf$ for illustration:
\begin{array}{|c|c|c|c|}
\hline
n& 2 & 3 & 4 & 5 & 6 & 7 & 8\\ \hline
χ_n & 1.6180 & 1.3247 & 1.2207 & 1.1673 & 1.1347 & 1.1128 & 1.0970 \\ \hline
\end{array}
Studied by Dutch architect Dom Hans van der Laan in the 1920s, they are basically the positive real solutions to the equations
\begin{align}
&x^n = x+1 \quad \quad \ (n\geq 2, \ n\in\mathbb{Z})\\
\end{align}
They appear to be quite nicely approximated by
\begin{align}
&\frac{2 n - 1 + \log 2}{2 n - 1 - \log 2}\\
\end{align}
I was just wondering whether anyone could shed some light on why this might be the case?
ref:
Plastic number: construction and applications
L Marohnić, T Strmečki
 A: Actually, there are only two morphic numbers. From this reference:  Jan Aarts, Robbert Fokkink, Godfried Kruijtzer, Morphic numbers, NAW 5/2 nr. 2 maart 2001, we have the definition
A real number $p$ > 1 is called a morphic number if
there exist natural numbers $k$ and $l$ such that
$$p + 1 = p^k \text{ and } \ p − 1 = p^{−l}$$ The only two numbers that satisfy this condition are the golden ratio and the plastic number.
You may be thinking of the Metallic mean, which are given by
$$m = \frac{n+\sqrt{n^2+4}}{2}, \quad n\ge 0$$
EDIT: The OP has pointed out that he meant the harmonious numbers. I'm familiar with these as well. They derive from the equation $X+1=X^m$, for $m>1$. In addition, they arise from the sequence
$$f_k=f_{k-m+1}+f_{k-m},\quad k<m\\
f_{k\le1}=1\\
\lim_{k\to\infty}\frac{f_{k+1}}{f_k}=X
$$
These constitute for about half of what I call the pseudomorphic numbers, the remainder of which are defined by $\chi-1=\chi^{-n}, n>1$, or
$$f_k=f_{k-1}+f_{k-1-n},\quad k>n+1\\
f_{k\le n+1}=1\\
\lim_{k\to\infty}\frac{f_{k+1}}{f_k}=\chi
$$
