63 lines
1.7 KiB
Markdown
63 lines
1.7 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.: $a(5) = a({101}_2) = 0$, $a(6) = a({110}_2) = 1$, $a(7) = a({111}_2) = 2$
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Define the sequence $b(n) = {(-1)}^{a(n)}$. This sequence is called the Rudin-Shapiro sequence.
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Also consider the summatory sequence of $b(n)$: $s(n) = \displaystyle\sum_{i = 0}^{n} b(i)$.
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The first couple of values of these sequences are:
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$$\begin{array}{lr} n & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\\\ a(n) & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 2 \\\\ b(n) & 1 & 1 & 1 & -1 & 1 & 1 & -1 & 1 \\\\ s(n) & 1 & 2 & 3 & 2 & 3 & 4 & 3 & 4 \end{array}$$
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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 $s(n)$.
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E.g.: $g(3, 3) = 6$, $g(4, 2) = 7$ and $g(54321, 12345) = 1\\,220\\,847\\,710$.
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Let $F(n)$ be the fibonacci sequence defined by:
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$$\begin{align} & F(0) = F(1) = 1 \text{ and} \\\\ & F(n) = F(n - 1) + F(n - 2) \text{ for } n > 1. \end{align}$$
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Define $GF(t) = g(F(t), F(t - 1))$.
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Find $\sum GF(t)$ for$ 2 ≤ t ≤ 45$.
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# --hints--
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`rudinShapiroSequence()` should return `3354706415856333000`.
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```js
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assert.strictEqual(rudinShapiroSequence(), 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 rudinShapiroSequence() {
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return true;
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}
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rudinShapiroSequence();
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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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