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For a positive integer n, let

$$R_n = \underbrace{111 \dots 1}_{n \text{ 1s}}.$$

For example, $$R_4 = 1111.$$

Find the polynomial p(x) such that $$p(R_n) = R_{2n + 1}$$ for all positive integers n. (Give your answer in base 10.)

Jun 6, 2020