Functions and Sequences Problem 
The function $F(k)$ is defined for positive integers as $F(1) = 1$,
  $F(2) = 1$, $F(3) = -1$ and $F(2k) = F(k)$, $F(2k + 1) = F(k)$ for $k \geq
 2$. Then $$F(1) + F(2) + \dotsb + F(63)$$ equals 
$\begin{array}{lr}
 (\text{A}) & 1 \\
 (\text{B}) & -1 \\
 (\text{C}) & -32 \\ 
 (\text{D}) & 32 \\
\end{array}$

My approach: $F(4)=1$, $F(5)=1$, $F(6)=-1$, $F(7)=-1$ (i.e., all values of $F$ are either $1$ or $-1$). I tried to find a pattern for which $F(x)$ repeats after a certain integer but tried till $F(30)$ and cannot find a solution.  Where am I going wrong?

We have $F(1)=1, F(2)=1, F(3)=-1$ and $F(2k)=F(k)$ and $F(2k+1)=F(k)$. 
We have to evaluate $S=\sum_{i=1}^{63}F(i)$
Notice that $S=F(1)+F(2)+F(3) + \sum_{i=2}^{31}[F(2k) + F(2k+1)]$
$\Rightarrow S=F(1)+F(2)+F(3) + \sum_{i=2}^{31}[2F(k)]$
$\Rightarrow S=1+2\sum_{i=2}^{31}F(k)$
$\Rightarrow S=1+2[F(2)+F(3)+\sum_{i=2}^{15}[F(2k)+F(2k+1)]]$
$\Rightarrow S=1+2\sum_{i=2}^{15}[F(2k)+F(2k+1)]$
$\Rightarrow S=1+4\sum_{i=2}^{15}F(k)$
$\Rightarrow S= 1+ 4[F(2)+F(3) + \sum_{i=2}^{7}[F(2k)+ F(2k+1)]]$
$\Rightarrow S = 1+ 8\sum_{i=2}^7 F(k)$
Similarly proceeding, we get 
$\Rightarrow S = 1+ 16\sum_{i=2}^3F(k)$
$\Rightarrow S =1$ 
