55 lines
1.3 KiB
Markdown
55 lines
1.3 KiB
Markdown
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---
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id: 5900f4ed1000cf542c50fffe
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title: 'Problem 384: Rudin-Shapiro sequence'
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challengeType: 5
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forumTopicId: 302048
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dashedName: problem-384-rudin-shapiro-sequence
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---
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# --description--
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Define the sequence a(n) as the number of adjacent pairs of ones in the binary expansion of n (possibly overlapping).
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E.g.: a(5) = a(1012) = 0, a(6) = a(1102) = 1, a(7) = a(1112) = 2
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Define the sequence b(n) = (-1)a(n). This sequence is called the Rudin-Shapiro sequence. Also consider the summatory sequence of b(n): .
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The first couple of values of these sequences are: n 0 1 2 3 4 5 6 7 a(n) 0 0 0 1 0 0 1 2 b(n) 1 1 1 -1 1 1 -1 1 s(n) 1 2 3 2 3 4 3 4
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The sequence s(n) has the remarkable property that all elements are positive and every positive integer k occurs exactly k times.
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Define g(t,c), with 1 ≤ c ≤ t, as the index in s(n) for which t occurs for the c'th time in s(n). E.g.: g(3,3) = 6, g(4,2) = 7 and g(54321,12345) = 1220847710.
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Let F(n) be the fibonacci sequence defined by: F(0)=F(1)=1 and F(n)=F(n-1)+F(n-2) for n>1.
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Define GF(t)=g(F(t),F(t-1)).
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Find ΣGF(t) for 2≤t≤45.
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# --hints--
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`euler384()` should return 3354706415856333000.
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```js
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assert.strictEqual(euler384(), 3354706415856333000);
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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 euler384() {
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return true;
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}
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euler384();
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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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