--description--

Consider the infinite polynomial series $A_G(x) = xG_1 + x^2G_2 + x^3G_3 + \cdots$, 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, $1, 4, 5, 9, 14, 23, \ldots$.

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.

--hints--

modifiedGoldenNuggets() should return 5673835352990

assert.strictEqual(modifiedGoldenNuggets(), 5673835352990);

--seed--

--seed-contents--

function modifiedGoldenNuggets() {

  return true;
}

modifiedGoldenNuggets();

--solutions--

// solution required