Weighted Mediants and Fractals

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.

22 pages, 11 figures

References in corpus (1)

• Regularity properties of the Stern enumeration of the rationals