# Problem 384: Rudin-Shapiro sequence

![graphic](img384.gif)

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.