Relations between $\pi$ and the golden ratio $\phi$ in the form of Bailey-Borwein-Plouffe-type formulas

TL;DR

This paper introduces a family of formulas expressing pi in terms of the golden ratio, extending Bailey-Borwein-Plouffe formulas and exploring connections with cyclotomic polynomials.

Contribution

It presents new Bailey-Borwein-Plouffe-type formulas linking pi and the golden ratio, with insights into cyclotomic polynomial connections.

Findings

New formulas expressing pi via the golden ratio

Connection established between these formulas and cyclotomic polynomials

Extension of Bailey-Borwein-Plouffe methodology

Abstract

We provide a family of expressions of $\pi$ in terms of the golden ratio $\phi$ in the same spirit of the formula obtained by Bailey, Borwein and Plouffe for $\pi$. Connection with cyclotomic polynomials is outlined.