$\pi$-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 $$\pi =\underset{n\to \infty }{\lim }{2}^{n+1}\cdot \sqrt{2-\underset{n}{\underbrace{\sqrt{2+\sqrt{2+\sqrt{2+...+\sqrt{2}}}}}}},$$ related to the smallest positive zero of $L_n$; the second is an exact formula for $\pi$ achieved thanks to some identities valid for $L_n$.

References in corpus (2)

• A Class of Periodic Continued Radicals
• Ordering of nested square roots of 2 according to Gray code