Modified Fibonacci golden nuggets

 Published on Saturday, 3rd February 2007, 07:00 am; Solved by 4192;
Difficulty rating: 55%

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.


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.