Periodicities which preserve and periodicities which destroy boundedness (Q609432)

In this nicely written paper the authors study the difference equation \[ x_{n+1}= \beta_n+{x_{n-2}\over x_n}, \quad n=0, 1, \dots \] They prove that every positive solution is bounded when \(\{\beta_n\}_{n=0}^\infty\) is a 2-period sequence of positive real numbers. They also show that there exist period-\(3m\) sequences \(\{\beta_n\}_{n=0}^\infty\) of positive real numbers such that the equation has unbounded solutions.