I'm trying to find the explicit form of the recursive function
$$P_{1(n)}=\left\lceil\frac12P_{1(n-1)}\right\rceil+\left\lfloor\frac12P_{2(n-1)}\right\rfloor\;.$$
First, let me explain what this says. This is a function that models a game in which three players split a pile of candy in any way they want to start. One player could even start with none. Each turn, players pass half of what they have to the left, keeping the extra candy if they have an odd amount.
So this formula gives the amount of can player 1 $(P_{1(n)})$ has after $n$ iterations. He keeps the ceiling of what he had cut in half and receives the floor of half of player 2's. Eventually the systems all reach a predictable amount based on the initial amount of candy, no matter how it is divided.
However, I am having a hard time describing this behavior in a matrix to find the explicit formula. Any suggestions? Is there a simpler way to approach this?