Weighted Mediants and Fractals

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.