Suppose that $\displaystyle S_n$ is the number of ways to seat $\displaystyle n$ women according to the rules.
Then $\displaystyle S_1=1,~S_2=2$ and if $\displaystyle n\ge 3$ then $\displaystyle S_n=S_{n-2}+S_{n-1}$
Can you explain why that works?
Hint: think about using strings already used.