## Rudin-Shapiro sequence

Published on Sunday, 13th May 2012, 02:00 am; Solved by 286;
Difficulty rating: 65%

### Problem 384

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): $s(n) = \sum \limits_{i = 0}^{n} {b(i)}$.

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 $\sum$ GF(t) for 2≤t≤45.