Modified Fibonacci golden nuggets

Consider the infinite polynomial series $A_G(x)=xG_1+x^2G_x+x^3G_3+\dots$, 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
$\frac{2}{5}$	2
$\frac{\sqrt{22}-2}{6}$	3
$\frac{\sqrt{137}-5}{14}$	4
$\frac{1}{2}$	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.