Modified Fibonacci golden nuggets (noch nicht ├╝bersetzt)

Problem 140

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 20th golden nugget is 211345365.

Find the sum of the first thirty golden nuggets.