2018-10-10 18:03:03 -04:00
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---
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id: 5900f3f51000cf542c50ff08
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2021-10-03 12:24:27 -07:00
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title: '问题 137:斐波那契金块'
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2018-10-10 18:03:03 -04:00
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challengeType: 5
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2021-02-06 04:42:36 +00:00
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forumTopicId: 301765
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2021-01-13 03:31:00 +01:00
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dashedName: problem-137-fibonacci-golden-nuggets
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2018-10-10 18:03:03 -04:00
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---
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2020-12-16 00:37:30 -07:00
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# --description--
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2018-10-10 18:03:03 -04:00
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2021-10-03 12:24:27 -07:00
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考虑无穷级数 $A_{F}(x) = xF_1 + x^2F_2 + x^3F_3 + \ldots$,其中 $F_k$ 是斐波那契数列 $1, 1, 2, 3, 5, 8, \ldots$ 的第 $k$ 项;即 $F_k = F_{k − 1} + F_{k − 2}, F_1 = 1$,$F_2 = 1$。
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2018-10-10 18:03:03 -04:00
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2021-10-03 12:24:27 -07:00
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在这个问题中,我们关注的是那些使得 $A_{F}(x)$ 为正整数的 $x$ 的值。
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2018-10-10 18:03:03 -04:00
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2021-10-03 12:24:27 -07:00
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令人惊讶的是:
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2018-10-10 18:03:03 -04:00
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2022-03-31 22:31:59 +05:30
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$$\begin{align} A_F(\frac{1}{2}) & = (\frac{1}{2}) × 1 + {(\frac{1}{2})}^2 × 1 + {(\frac{1}{2})}^3 × 2 + {(\frac{1}{2})}^4 × 3 + {(\frac{1}{2})}^5 × 5 + \cdots \\\\
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& = \frac{1}{2} + \frac{1}{4} + \frac{2}{8} + \frac{3}{16} + \frac{5}{32} + \cdots \\\\ & = 2 \end{align}$$
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2018-10-10 18:03:03 -04:00
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2021-10-03 12:24:27 -07:00
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前五个对应的自然数 $x$ 如下。
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2021-02-06 04:42:36 +00:00
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2021-10-03 12:24:27 -07:00
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| $x$ | $A_F(x)$ |
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| --------------------------- | -------- |
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| $\sqrt{2} − 1$ | $1$ |
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| $\frac{1}{2}$ | $2$ |
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| $\frac{\sqrt{13} − 2}{3}$ | $3$ |
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| $\frac{\sqrt{89} − 5}{8}$ | $4$ |
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| $\frac{\sqrt{34} − 3}{5}$ | $5$ |
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2021-02-06 04:42:36 +00:00
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2021-10-03 12:24:27 -07:00
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当 $x$ 是有理数时,我们称 $A_F(x)$ 是一个金砖,因为这样的数字逐渐变得稀少;例如,第 10 个金砖是 74049690。
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2021-02-06 04:42:36 +00:00
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2021-10-03 12:24:27 -07:00
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请求出第 15 个金砖。
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2018-10-10 18:03:03 -04:00
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2020-12-16 00:37:30 -07:00
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# --hints--
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2018-10-10 18:03:03 -04:00
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2021-10-03 12:24:27 -07:00
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`goldenNugget()` 应该返回 `1120149658760`。
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2018-10-10 18:03:03 -04:00
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```js
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assert.strictEqual(goldenNugget(), 1120149658760);
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2018-10-10 18:03:03 -04:00
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```
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2021-01-13 03:31:00 +01:00
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# --seed--
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## --seed-contents--
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```js
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function goldenNugget() {
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return true;
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}
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2021-10-03 12:24:27 -07:00
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goldenNugget();
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2021-01-13 03:31:00 +01:00
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```
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2020-12-16 00:37:30 -07:00
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# --solutions--
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2020-08-13 17:24:35 +02:00
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2021-01-13 03:31:00 +01:00
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```js
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// solution required
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```
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