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arXiv · 2608.12739

On the Lorenz-Fibonacci sequences and substitutions

Bernardo San Martín·Víctor Sirvent
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Abstract

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 $k\geq 2r+1$, and $k\geq 3$. 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.

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Bernardo San Martín, Víctor Sirvent. 2026-08-13. On the Lorenz-Fibonacci sequences and substitutions. https://arxiv.org/abs/2608.12739

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