Solution :
Put and . By using of trigonometric formulas
and taking into account that and , we have and .
Therefore, is integer. If we define (mod 5) and (mod 5), then we have and modulo 5. Obviously, is uniquely determined by initial values and and is a periodic sequence.
If then , , , ,
If then , , , ,
(Both sequences have a period 4). Finally, a = 1 or 4.