# This looks awfully familiar

Calculus Level 5

Define a sequence of polynomials $$\lbrace T_{n} \rbrace_{n=1}$$ in $$x$$ by $$T_{1} = x$$, $$T_{2} = 2x^{2}-1$$ and $$2x \ T_{n} = T_{n+1}+T_{n-1}$$ where $$n \geq 2$$.

Let $$S$$ be the set of all $$n$$ for which there exists a polynomial function $$f$$ such that $$T_{n} = f \circ f$$, and $$T_{n}(\frac{\sqrt{3}}{2})$$ is locally maximized.

Find the value of $$\displaystyle \left\lfloor 1000 \cdot \sum_{n \in S} \frac{1}{n} \right \rfloor$$.

Note: $$(f \circ g)(x) = f(g(x))$$.

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