关于洛伦兹-斐波那契序列与代换
On the Lorenz-Fibonacci sequences and substitutions
AI总结：
本文研究两类$(k,r)$-洛伦兹-斐波那契序列，引入对应代换并探究其组合性质，该序列与$k$阶斐波那契序列相关，源于洛伦兹吸引子动力学研究。
AI中文摘要：
本文研究两类整数序列：第一类和第二类$(k,r)$-洛伦兹-斐波那契序列，其特征多项式为$p_{k,r}(x)=x^k-x^{k-1}-\cdots-x^r+x^{r-1}+\cdots+x+1$，其中$k\geq2r+1$且$k\geq3$。当$r=0$时，这些序列包含著名的$k$阶斐波那契序列。它们自然产生于洛伦兹吸引子的动力学研究中。我们引入两类代换（基于$k$个符号的字母表），使其与每类整数序列关联并共享相同的多项式，研究这些代换的主要组合性质。
英文摘要：
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.
