 Solution

The first characteristic polynomial factors asThe root condition holds precisely for $0<\alpha <2$: in this range the two reciprocal roots are distinct and on the unit circle; at either endpoint a unit root is repeated, and outside it one root has modulus greater than one. Consistency is given, so these and only these values are convergent. The exceptional order-three value $\alpha =6$ is outside this interval; consequently every convergent case has order two.