47fc3c6761
* fix: clean-up Project Euler 281-300 * fix: missing image extension * fix: missing power Co-authored-by: Tom <20648924+moT01@users.noreply.github.com> * fix: missing subscript Co-authored-by: Tom <20648924+moT01@users.noreply.github.com> Co-authored-by: Tom <20648924+moT01@users.noreply.github.com>
53 lines
1.1 KiB
Markdown
53 lines
1.1 KiB
Markdown
---
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id: 5900f4951000cf542c50ffa8
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title: 'Problem 297: Zeckendorf Representation'
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challengeType: 5
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forumTopicId: 301949
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dashedName: problem-297-zeckendorf-representation
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---
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# --description--
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Each new term in the Fibonacci sequence is generated by adding the previous two terms.
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Starting with 1 and 2, the first 10 terms will be: 1, 2, 3, 5, 8, 13, 21, 34, 55, 89.
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Every positive integer can be uniquely written as a sum of nonconsecutive terms of the Fibonacci sequence. For example, 100 = 3 + 8 + 89.
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Such a sum is called the Zeckendorf representation of the number.
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For any integer $n>0$, let $z(n)$ be the number of terms in the Zeckendorf representation of $n$.
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Thus, $z(5) = 1$, $z(14) = 2$, $z(100) = 3$ etc.
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Also, for $0 < n < {10}^6$, $\sum z(n) = 7\\,894\\,453$.
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Find $\sum z(n)$ for $0 < n < {10}^{17}$.
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# --hints--
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`zeckendorfRepresentation()` should return `2252639041804718000`.
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```js
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assert.strictEqual(zeckendorfRepresentation(), 2252639041804718000);
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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 zeckendorfRepresentation() {
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return true;
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}
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zeckendorfRepresentation();
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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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