On the Lorenz-Fibonacci sequences and substitutions

In the present article, we consider two families of integer sequences, the $(k,r)$-Lorenz-Fibonacci sequences of the first and second kind, whose characteristic polynomial is $$p_{k,r}(x)=x^k-x^{k-1}-\cdots -x^r+x^{r-1}+\cdots +x+1,$$ where $k\ge 2r+1$, and $k\ge 3$. These families include the well-known $k$-bonacci sequence, when $r=0$. These sequences arise naturally in the study of the dynamics of the Lorenz attractor. We introduce two families of substitutions (in an alphabet of $k$ symbols) so that they are associated with each of the families of integer sequences and share the same polynomial. We study the main combinatorial properties of these substitutions.