Two species of particles, $X$ and $Y$, interact recursively. In each step, every particle of type $X$ splits into two of type $X$ and one of type $Y$. Every particle of type $Y$ converts into one of type $X$. If we start with 1 particle of type $X$ and 0 of type $Y$, let $X_n$ and $Y_n$ be the populations after $n$ steps. Which of the following is the correct recurrence relation for $X_n$?