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?

With induction it can be shown that:$$F(2^k)+\cdots+F(2^{k+1}-1)=0$$
for $k=1,2,\dots$
This because $F(2^1)+F(2^2-1)=F(2)+F(3)=1+(-1)=0$ and:$$F(2^{k+1})+F(2^{k+1}+1)+\cdots+F(2^{k+2}-2)+F(2^{k+2}-1)=2F(2^k)+\cdots+2F(2^{k+1}-1)$$
Then $$F(1)+\cdots+F(63)=F(1)+\sum_{k=1}^5\left[F(2^k)+\cdots+F(2^{k+1}-1)\right]=F(1)=1$$
