Algebra Level 5

Denote by $$F_0(x), F_1(x), ...$$ the sequence of Fibonacci polynomials, which satisfy the recurrence $$F_0(x) = 1, F_1(x) = x,$$ and $$F_n(x) = xF_{n−1}(x) + F_{n−2}(x)$$ for all $$n ≥ 2$$. It is given that there exist unique integers $$λ_0, λ_1, . . ., λ_{1000}$$ such that $$x^{1000} = \sum_{i=0}^{1000} λ_iF_i(x)$$ for all real $$x$$. For which integer $$k$$ is $$|λ_k|$$ maximized?

Note: Full credit is given to the makers of this problem.

×