# 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: \((f \circ g)(x) = f(g(x))\).