$\pi$-formulas and Gray code

Abstract

In previous papers we introduced a class of polynomials which follow the same recursive formula as the Lucas-Lehmer numbers, studying the distribution of their zeros and remarking that this distribution follows a sequence related to the binary Gray code. It allowed us to give an order for all the zeros of every polynomial $L_n$. In this paper, the zeros, expressed in terms of nested radicals, are used to obtain two formulas for $\pi$: the first can be seen as a generalization of the known formula $$\pi=\lim_{n\rightarrow \infty} 2^{n+1}\cdot \sqrt{2-\underbrace{\sqrt{2+\sqrt{2+\sqrt{2+...+\sqrt{2}}}}}_{n}} \ ,$$ related to the smallest positive zero of $L_n$; the second is an exact formula for $\pi$ achieved thanks to some identities valid for $L_n$.