arXiv subjects

Huang Yuke

Publications and source records attributed to Huang Yuke.

3 recordsLinked to original sources

arXiv2016

Envelope Words and Return Words Sequences in the Period-doubling Sequence

We consider the infinite one-sided sequence generated by the period-doubling substitution $σ(a,b)=(ab,aa)$, denoted by $\mathbb{D}$. Since $\mathbb{D}$ is uniformly recurrent, each factor $ω$ appears infinite many times in the sequence, which is arranged as $ω_p$ $(p\ge 1)$. Let $r_p(ω)$ be the $p$-th return word over $ω$. The main result is: for each factor $ω$, the sequence $\{r_p(ω)\}_{p\geq1}$ is $Θ_1$ or $Θ_2$, which are substitutive sequences and determined completely in this paper.

math.DS↗

arXiv2016

The numbers of distinct and repeated squares and cubes in the Tribonacci sequence

The Tribonacci sequence $\mathbb{T}$ is the fixed point of the substitution $σ(a,b,c)=(ab,ac,a)$. The main result is twofold: (1) we give the explicit expressions of the numbers of distinct squares and cubes in $\mathbb{T}[1,n]$ (the prefix of $\mathbb{T}$ of length $n$); (2) we give algorithms for counting the number of repeated squares and cubes in $\mathbb{T}[1,n]$ for all $n$; then get explicit expressions for some special $n$ such as $n=t_m$ (the Tribonacci number).

math.DS↗

arXiv2016

The numbers of repeated palindromes in the Fibonacci and Tribonacci sequences

The Fibonacci sequence $\mathbb{F}$ is the fixed point beginning with $a$ of morphism $σ(a,b)=(ab,a)$. Since $\mathbb{F}$ is uniformly recurrent, each factor $ω$ appears infinite many times in the sequence which is arranged as $ω_p$ $(p\ge 1)$. Here we distinguish $ω_p\neqω_q$ if $p\neq q$. In this paper, we give algorithm for counting the number of repeated palindromes in $\mathbb{F}[1,n]$ (the prefix of $\mathbb{F}$ of length $n$). That is the number of the pairs $(ω, p)$, where $ω$ is a palindrome and $ω_p\prec\mathbb{F}[1,n]$. We also get explicit expressions for some special $n$ such as $n=f_m$ (the $m$-th Fibonacci number). The similar results are also given to the Tribonacci sequence, the fixed point beginning with $a$ of morphism $τ(a,b,c)=(ab,ac,a)$.

math.DS↗