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id	title	challengeType	forumTopicId	dashedName
5900f3fa1000cf542c50ff0c	Problem 140: Modified Fibonacci golden nuggets	5	301769	problem-140-modified-fibonacci-golden-nuggets

--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.

`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 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

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