# Annoying Reciprocals

Calculus Level 3

If $$x$$ is a complex number satisfying $${ x }^{ 2 }+x+1=0$$, then for all positive integers $$n$$: $\sum _{k=1}^n \left(x^k+\frac 1{x^k} \right)^2 =n+a \left \lfloor \frac nb \right \rfloor$

where $$a$$ and $$b$$ are positive integers. Find $$a+b$$.

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