1970 AMC 12 Question 16

If $F(n)$ is a function such that $F(1)=F(2)=F(3)=1,$ and such that $$
F(n+1)=\frac{F(n)F(n-1)+1}{F(n-2)}
$$ for $n\ge3,$ then $F(6)$ is equal to: