$4$ people, designated Black, White, Red, and Yellow, are playing a game. Each of them has unlimited number of balls, all of that person’s designated color. They work together to build a line of balls. Each player can either add one colored ball or throw away the last ball in line and add two colored balls. The game begins with a single green ball.
Find $a_n$ (recursion) if the players plays in a fixed rotation. Then find another $a_n$ that the players can play in any order, provided that no player ever makes two moves in a row.
I'm struggling finding the first $a_n$, anyone has any suggestions?
I thought about the situation if the turns are $B\to W\to R\to Y$. For $B$ we will have two options, and both of them will end with $2$ balls in the line, which means we can finish with $a_{n-2}$ possibilities. With White we will have $3$ balls or $1$ ball in line. With Red we will have $4$ balls or $2$ balls in line if it was $3$ balls after White played, and if we had one ball after White played, we will have $3$ balls or $2$ balls. And there will be $8$ options after Yellow plays. And then we sum all the possibilities into a huge, possibly wrong, equation.