## Rudin-Shapiro sequence

### Problem 384

Published on Sunday, 13th May 2012, 02:00 am; Solved by 232; Difficulty rating: 65%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): .

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.