freeCodeCamp/curriculum/challenges/english/10-coding-interview-prep/project-euler/problem-384-rudin-shapiro-sequence.md

id, title, challengeType, forumTopicId, dashedName

id	title	challengeType	forumTopicId	dashedName
5900f4ed1000cf542c50fffe	Problem 384: Rudin-Shapiro sequence	5	302048	problem-384-rudin-shapiro-sequence

--description--

Define the sequence a(n) as the number of adjacent pairs of ones in the binary expansion of n (possibly overlapping).

E.g.: a(5) = a({101}_2) = 0, a(6) = a({110}_2) = 1, a(7) = a({111}_2) = 2

Define the sequence b(n) = {(-1)}^{a(n)}. This sequence is called the Rudin-Shapiro sequence.

Also consider the summatory sequence of b(n): s(n) = \displaystyle\sum_{i = 0}^{n} b(i).

The first couple of values of these sequences are:

$$\begin{array}{lr} n & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ a(n) & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 2 \\ b(n) & 1 & 1 & 1 & -1 & 1 & 1 & -1 & 1 \\ s(n) & 1 & 2 & 3 & 2 & 3 & 4 & 3 & 4 \end{array}$$

The sequence s(n) has the remarkable property that all elements are positive and every positive integer k occurs exactly k times.

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) = 1\\,220\\,847\\,710.

Let F(n) be the fibonacci sequence defined by:

$$\begin{align} & F(0) = F(1) = 1 \text{ and} \\ & F(n) = F(n - 1) + F(n - 2) \text{ for } n > 1. \end{align}$$

Define GF(t) = g(F(t), F(t - 1)).

Find \sum GF(t) for$ 2 ≤ t ≤ 45$.

--hints--

rudinShapiroSequence() should return 3354706415856333000.

assert.strictEqual(rudinShapiroSequence(), 3354706415856333000);

--seed--

--seed-contents--

function rudinShapiroSequence() {

  return true;
}

rudinShapiroSequence();

--solutions--

// solution required