If x² + x +1 =0, then find the value of:
(x+1/x)² + (x² + x^-2)² + ...+ (x⁹ + x^-9)²

(In reply to computer solution (spoiler) by Charlie)

The behavior you noted happens because the two solutions are two of the three complex cube roots of one. 
x=cos(120)+i*sin(120) and x=cos(240)+i*sin(240).

So if you square either of them you get the other, but cubing gives 1.

Posted by Jer on 2015-01-14 11:26:08