Yatir from Israel wrote this article on numbers that can be written as $ 2^n-n $ where n is a positive integer.

A sequence of polynomials starts 0, 1 and each poly is given by combining the two polys in the sequence just before it. Investigate and prove results about the roots of the polys.

Challenge Level

Let $$ f_{0}(x)= \frac{1}{1-x}$$ and $$f_{n}(x)=f_{0}(f_{n-1}(x))$$
where $n = 1,2, 3, 4, ...$

Evaluate $f_{2000}(2000)$

Evaluate $f_{2000}(2000)$

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