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# Problem 384

Rudin-Shapiro sequence

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

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 ΣGF(t) for 2≤t≤45.

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

F(0)=F(1)=1 且