Weighted Mediants and Fractals

TL;DR

This paper explores a generalization of Stern-Brocot sequences through weighted mediants, focusing on the case k=3, and reveals fractal-like properties in the cross-differences of these sequences.

Contribution

It introduces and analyzes the properties of weighted mediants in Stern-Brocot sequences, especially for k=3, and proves their fractal-like cross-difference patterns.

Findings

Cross-differences exhibit fractal-like patterns.

Properties of weighted mediants are characterized for k=3.

Sequence behavior generalizes classical Stern-Brocot properties.

Abstract

In this paper we study a natural generalization of the Stern-Brocot sequences which comes from the introduction of weighted mediants. We focus our attention on the case $k = 3$, in which $(2a + c)/(2b + d)$ and $(a + 2c)/(b + 2d)$ are the two mediants inserted between $a/b$ and $c/d$. We state and prove several properties about the cross-differences of Stern-Brocot sequences with $k = 3$, and give a proof of the fractal-like rule that describes the cross-differences of the unit $k = 3$ Stern-Brocot sequences, i.e. the one with usual starting terms $0/1, 1/1$ and with reduction of fractions.