Why are the Fibonacci quotients $\frac{F_{2n}}{F_{2n+1}}$ the only rationals that make $\frac{x}{1 - x - x^2}$ a non-negative integer? I've been studying A081018 as part of a programming challenge. Skipping some of the introductory material, the crux of the question is that we have the equation
$$S = \frac{x}{1 - x - x^2}$$
and we want to know which values $x \in \mathbb{Q}$ produce nonnegative integer values of $S$.
A081018 (linked above) asserts that all solutions to this problem are of the form $x = \frac{F_{2n}}{F_{2n+1}}$, resulting in $S = F_{2n} F_{2n+1}$ (where $F_n$ is the $n$th Fibonacci number, with $F_0 = 0, F_1 = 1$).
Now, I understand why those $x$ values are solutions to the problem. As a very quick summary, we have
$$S = \frac{\frac{F_{2n}}{F_{2n+1}}}{1 - \frac{F_{2n}}{F_{2n+1}} - \frac{F_{2n}^2}{F_{2n+1}^2}} = \frac{F_{2n}F_{2n+1}}{F_{2n+1}(F_{2n+1} - F_{2n}) - F_{2n}^2} = \frac{F_{2n}F_{2n+1}}{F_{2n+1}F_{2n-1} - F_{2n}^2} = (-1)^{2n} F_{2n} F_{2n+1} = F_{2n} F_{2n+1}$$
Where the penultimate equality is due to the Cassini identity.
However, I'm struggling to see why these are the only solutions. I've tried some basic analysis with the norm in $\mathbb{Q}(\sqrt{5})$ (since the Fibonacci numbers seems to have a particular attachment to $\sqrt{5}$) but came up mostly fruitless. How can we demonstrate that every $x \in \mathbb{Q}$ which produces a nonnegative integer $S$ must be of the form $x = \frac{F_{2n}}{F_{2n+1}}$?
 A: Here are other questions where the Conway Topograph was relevant, in several I posted a diagram.
Producing the diagram is algorithmic in some ways. I have simple programs  to get the numbers right. Professors have asked about software to make these, I don't see why not. It is the final graphics that takes more  background than I have.
At the end I also list four books that discuss the topograph. The book by Hatcher is most similar to my choices; on the whole, I put in more information in one diagram than these authors.
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http://www.maths.ed.ac.uk/~aar/papers/conwaysens.pdf   (Conway)
https://www.math.cornell.edu/~hatcher/TN/TNbook.pdf    (Hatcher)
http://bookstore.ams.org/mbk-105/                  (Weissman)
http://www.springer.com/us/book/9780387955872             (Stillwell)
