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$a_{k}=x^k+\frac{1}{x^k}=(x^k+\frac{1}{x^k})*1=(x^k+\frac{1}{x^k})*-(x +\frac{1}{x})=-(a_{k+1}+a_{k-1}) \\ a_{k+1} = -(a_k+a_{k-1}) \\ a_{k} = -(a_{k-1} + a_{k-2}) \\ a_{k+1} = a_{k-2}$
以3为周期的数列