On a BBP-type formula for $\pi^2$ in the golden ratio base

TL;DR

This paper provides a detailed proof of a BBP-type formula for pi squared in the golden ratio base, and extends the methodology to derive a new efficient formula for zeta(3).

Contribution

It introduces a new proof of a BBP-type formula for pi squared in the golden ratio base and extends the approach to zeta(3) with a Machin-like formula.

Findings

Established a BBP-type formula for pi^2 in the golden ratio base.

Extended the methodology to derive a new Machin-like formula for zeta(3).

Demonstrated the geometric identity linking phi to roots of unity.

Abstract

This paper presents a detailed, self-contained proof of a BBP-type formula for $\pi^2$ expressed in the golden ratio base, $\phi$. The formula was discovered empirically by the author in 2004. The proof presented herein is built upon a fundamental geometric identity connecting $\phi$ to the fifth roots of unity, offering an intuitive and direct path to the result. The power of the underlying methodology is then demonstrated by extending it to establish a new, computationally efficient Machin-like formula for $\zeta(3)$, expressed through rapidly converging, hierarchical series involving the golden ratio.