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* fix: clean-up Project Euler 121-140 * fix: corrections from review Co-authored-by: Sem Bauke <46919888+Sembauke@users.noreply.github.com> * fix: missing backticks Co-authored-by: Kristofer Koishigawa <scissorsneedfoodtoo@gmail.com> * fix: corrections from review Co-authored-by: Tom <20648924+moT01@users.noreply.github.com> * fix: missing delimiter Co-authored-by: Sem Bauke <46919888+Sembauke@users.noreply.github.com> Co-authored-by: Kristofer Koishigawa <scissorsneedfoodtoo@gmail.com> Co-authored-by: Tom <20648924+moT01@users.noreply.github.com>
53 lines
1.4 KiB
Markdown
53 lines
1.4 KiB
Markdown
---
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id: 5900f3fa1000cf542c50ff0c
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title: 'Problem 140: Modified Fibonacci golden nuggets'
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challengeType: 5
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forumTopicId: 301769
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dashedName: problem-140-modified-fibonacci-golden-nuggets
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---
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# --description--
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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, $1, 4, 5, 9, 14, 23, \ldots$.
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For this problem we shall be concerned with values of $x$ for which $A_G(x)$ is a positive integer.
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The corresponding values of $x$ for the first five natural numbers are shown below.
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| $x$ | $A_G(x)$ |
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|-----------------------------|----------|
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| $\frac{\sqrt{5} − 1}{4}$ | $1$ |
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| $\frac{2}{5}$ | $2$ |
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| $\frac{\sqrt{22} − 2}{6}$ | $3$ |
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| $\frac{\sqrt{137} − 5}{14}$ | $4$ |
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| $\frac{1}{2}$ | $5$ |
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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.
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# --hints--
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`modifiedGoldenNuggets()` should return `5673835352990`
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```js
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assert.strictEqual(modifiedGoldenNuggets(), 5673835352990);
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```
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# --seed--
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## --seed-contents--
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```js
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function modifiedGoldenNuggets() {
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return true;
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}
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modifiedGoldenNuggets();
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```
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# --solutions--
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```js
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// solution required
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```
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