The Balkans Continued Fraction

TL;DR

This paper introduces generalized continued fractions related to Catalan's constant, categorized into regions called 'The Balkans,' supported by extensive numerical testing but lacking formal proofs.

Contribution

It provides new, numerically validated formulae for continued fractions involving Catalan's constant, expanding understanding without formal proofs.

Findings

Numerical evidence supports the proposed continued fraction formulae.

Different regions in the plane correspond to distinct types of continued fractions.

The formulas are not formally proven but are extensively tested numerically.

Abstract

In a previous escapade we gave a collection of continued fractions involving Catalan's constant. This paper provides more general formulae governing those continued fractions. Having distinguished different cases associated to regions in the plan, we nickname those continued fractions \enquote{The Balkans} as they divide into areas which are related but still different in nature. Because we do not provide formal proofs of those machine-constructed formulae we do not claim them to be theorems. Still, each and every proposed formula was extensively tested numerically.