Fibonacci Golden Nuggets
Consider the infinite polynomial series $A_F(x) = x F_1 + x^2 F_2 + x^3 F_3 + \dots$, where $F_k$ is the $k$th term in the Fibonacci sequence: $1, 1, 2, 3, 5, 8, \dots$; that is, $F_k = F_{k-1} + F_{k-2}$, $F_1 = 1$ and $F_2 = 1$.
For this problem we shall be interested in values of $x$ for which $A_F(x)$ is a positive integer.
| Surprisingly | $\begin{align*} A_F(\tfrac{1}{2}) &= (\tfrac{1}{2})\times 1 + (\tfrac{1}{2})^2\times 1 + (\tfrac{1}{2})^3\times 2 + (\tfrac{1}{2})^4\times 3 + (\tfrac{1}{2})^5\times 5 + \cdots \\ &= \tfrac{1}{2} + \tfrac{1}{4} + \tfrac{2}{8} + \tfrac{3}{16} + \tfrac{5}{32} + \cdots \\ &= 2 \end{align*}$ |
The corresponding values of $x$ for the first five natural numbers are shown below.
| $x$ | $A_F(x)$ |
|---|---|
| $\sqrt{2}-1$ | $1$ |
| $\tfrac{1}{2}$ | $2$ |
| $\frac{\sqrt{13}-2}{3}$ | $3$ |
| $\frac{\sqrt{89}-5}{8}$ | $4$ |
| $\frac{\sqrt{34}-3}{5}$ | $5$ |
We shall call $A_F(x)$ a golden nugget if $x$ is rational, because they become increasingly rarer; for example, the $10$th golden nugget is $74049690$.
Find the $15$th golden nugget.