53 lines
1.4 KiB
Markdown
53 lines
1.4 KiB
Markdown
---
|
||
id: 5900f3fa1000cf542c50ff0c
|
||
title: 'Problem 140: Modified Fibonacci golden nuggets'
|
||
challengeType: 5
|
||
forumTopicId: 301769
|
||
dashedName: 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`
|
||
|
||
```js
|
||
assert.strictEqual(modifiedGoldenNuggets(), 5673835352990);
|
||
```
|
||
|
||
# --seed--
|
||
|
||
## --seed-contents--
|
||
|
||
```js
|
||
function modifiedGoldenNuggets() {
|
||
|
||
return true;
|
||
}
|
||
|
||
modifiedGoldenNuggets();
|
||
```
|
||
|
||
# --solutions--
|
||
|
||
```js
|
||
// solution required
|
||
```
|