Taking these type of problems to a new level

Algebra Level 4

Suppose that a real number xx satisfies the equation

x2+1x2=3x^2+\frac{1}{x^2}=3

Define a sequence ana_{n} as an=xn+1xna_{n}=x^n+\frac{1}{x^n}. How many of a1,a2,a1000a_{1}, a_{2}, \cdots a_{1000} are integers?

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