A pair of positive integers $x_0$ and $y_0$ are selected and further pairs are generated according to the following rule:
After the values of $x_n$ and $y_n$ have been found, a coin is tossed. If it lands heads, then $x_{n+1} = x_n −1$ and $y_{n+1} = x_n$. Otherwise, if it lands tails, then $x_{n+1} = y_n − 2$ and $y_{n+1} = x_n + 1$.
The process ends when a non-positive value of $x$ or $y$ is generated. Is there a choice of initial values for $x_0$ and $y_0$ and sequence of coin toss outcomes for which the sequence of $x$’s and $y$’s does not terminate?