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id	challengeType	title	forumTopicId
5900f4ed1000cf542c50fffe	5	Problem 384: Rudin-Shapiro sequence	302048

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(1012) = 0, a(6) = a(1102) = 1, a(7) = a(1112) = 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): .

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

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) = 1220847710.

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.

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

Find ΣGF(t) for 2≤t≤45.

Instructions

Tests

tests:
  - text: <code>euler384()</code> should return 3354706415856333000.
    testString: assert.strictEqual(euler384(), 3354706415856333000);

Challenge Seed

function euler384() {
  // Good luck!
  return true;
}

euler384();

Solution

// solution required

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