= Solution
The first characteristic <polynomial> factors as
$$
\rho(z)=z^3+(2\alpha-3)(z^2-z)-1
=(z-1)\{z^2+(2\alpha-2)z+1\}.
$$
The 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.
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