Family size making two gender events independent

A family has $n\ge2$ children, each independently a boy or a girl with probability $\tfrac12$. Let $A$ be the event that both sexes are represented among the children, and let $B$ be the event that there is no more than one girl. For which single value of $n$ are $A$ and $B$ independent?

$n=3$. $P(A)=1-2^{1-n}$, $P(B)=\tfrac{n+1}{2^n}$, and $A\cap B$ means exactly one girl, $P=\tfrac n{2^n}$. Independence requires $n=(n+1)(1-2^{1-n})$, i.e. $2^{n-1}=n+1$, whose only solution is $n=3$.