1.7 KiB
1.7 KiB
| id | title | challengeType | forumTopicId | dashedName |
|---|---|---|---|---|
| 5900f3f51000cf542c50ff08 | Problem 137: Fibonacci golden nuggets | 5 | 301765 | problem-137-fibonacci-golden-nuggets |
--description--
Consider the infinite polynomial series A_{F}(x) = xF_1 + x^2F_2 + x^3F_3 + \ldots, where F_k is the k$th term in the Fibonacci sequence: $1, 1, 2, 3, 5, 8, \ldots; that is, F_k = F_{k − 1} + F_{k − 2}, F_1 = 1 and F_2 = 1.
For this problem we shall be interested in values of x for which A_{F}(x) is a positive integer.
Surprisingly
\begin{align}
A_F(\frac{1}{2}) & = (\frac{1}{2}) × 1 + {(\frac{1}{2})}^2 × 1 + {(\frac{1}{2})}^3 × 2 + {(\frac{1}{2})}^4 × 3 + {(\frac{1}{2})}^5 × 5 + \cdots \\\\
& = \frac{1}{2} + \frac{1}{4} + \frac{2}{8} + \frac{3}{16} + \frac{5}{32} + \cdots \\\\
& = 2
\end{align}$$
The corresponding values of $x$ for the first five natural numbers are shown below.
| $x$ | $A_F(x)$ |
|---------------------------|----------|
| $\sqrt{2} − 1$ | $1$ |
| $\frac{1}{2}$ | $2$ |
| $\frac{\sqrt{13} − 2}{3}$ | $3$ |
| $\frac{\sqrt{89} − 5}{8}$ | $4$ |
| $\frac{\sqrt{34} − 3}{5}$ | $5$ |
We shall call $A_F(x)$ a golden nugget if $x$ is rational, because they become increasingly rarer; for example, the 10th golden nugget is 74049690.
Find the 15th golden nugget.
# --hints--
`goldenNugget()` should return `1120149658760`.
```js
assert.strictEqual(goldenNugget(), 1120149658760);
```
# --seed--
## --seed-contents--
```js
function goldenNugget() {
return true;
}
goldenNugget();
```
# --solutions--
```js
// solution required
```