Eccoci.

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

The first couple of values of these sequences are:

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 $1\le c\le t$, 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 $2\le t\le 45$.