Modified Fibonacci Golden Nuggets
Consider the infinite polynomial series $A_G(x) = x G_1 + x^2 G_2 + x^3 G_3 + \cdots$, where $G_k$ is the $k$th term of the second order recurrence relation $G_k = G_{k-1} + G_{k-2}$, $G_1 = 1$ and $G_2 = 4$; that is, $1, 4, 5, 9, 14, 23, \dots$.
For this problem we shall be concerned with values of $x$ for which $A_G(x)$ is a positive integer.
The corresponding values of $x$ for the first five natural numbers are shown below.
| $x$ | $A_G(x)$ |
|---|---|
| $\frac{\sqrt{5}-1}{4}$ | $1$ |
| $\tfrac{2}{5}$ | $2$ |
| $\frac{\sqrt{22}-2}{6}$ | $3$ |
| $\frac{\sqrt{137}-5}{14}$ | $4$ |
| $\tfrac{1}{2}$ | $5$ |
We shall call $A_G(x)$ a golden nugget if $x$ is rational, because they become increasingly rarer; for example, the $20$th golden nugget is $211345365$.
Find the sum of the first thirty golden nuggets.