Let me take any arbitrary number and try to find what value it gets mapped under function F.
For n=100 (say) what is $F(100)$ ?
$F(100)=F(50)=F(25)=F(12)=F(6)=F(3)=-1$. Finally $F(100)$ becomes $-1$. But it's not a good idea to start with higher values as it leads to many calls to F. If we start with lower values then you will see that our computation overhead will be very less.
(It is like using Dynamic Programming in Bottom Up fashion. If you don't know dynamic programming then No problem, It is not required at all, Continue further.)
Observe that what $F(2)$ generates? (since recursive equation is defined for $k\geq2$ so I can not put k=1 there. I am starting with k=2 and k=3)
$F(2)=F(4)$ and $F(2)= F(5)$.
Similarly see, what $F(3)$ generates?. $F(3) =F(6)$ and $F(3)=F(7)$.
In this way, we can make a tree and this problem is done. :)
- $\begin{align*} &\sum_{k=1}^{100}F(k) = 32 - 5 + 1 = \bf \large \color{red}28 \\ \end{align*}$