Some Properties of the $k$-bonacci Words on Infinite Alphabet

Abstract

The Fibonacci word $W$ on an infinite alphabet was introduced in [Zhang et al., Electronic J. Combinatorics 2017 24(2), 2-52] as a fixed point of the morphism $2i\rightarrow (2i)(2i+1)$, $(2i+1) \rightarrow (2i+2)$, $i\geq 0$.
 Here, for any integer $k>2$, we define the infinite $k$-bonacci word $W^{(k)}$ on the infinite alphabet as $\varphi_k^{\omega}(0)$, where the morphism $\varphi_k$ on the alphabet $\mathbb{N}$ is defined for any $i\geq 0$ and any $0\leq j\leq k-1$, by
 \begin{equation*}
 \varphi_k(ki+j) = \left\{
 \begin{array}{ll}
 (ki)(ki+j+1) & \text{if } j = 0,\cdots ,k-2,\\
 (ki+j+1)& \text{otherwise}.
 \end{array} \right.
 \end{equation*}
 We consider the sequence of finite words $(W^{(k)}_n)_{n\geq 0}$, where $W^{(k)}_n$ is the prefix of $W^{(k)}$ whose length is the $(n+k)$-th $k$-bonacci number. We then provide a recursive formula for the number of palindromes that occur in different positions of $W^{(k)}_n$. Finally, we obtain the structure of all palindromes occurring in $W^{(k)}$ and based on this, we compute the palindrome complexity of $W^{(k)}$, for any $k>2$.

Kernel Words and Gap Sequences of the Tribonacci Word on an Infinite Alphabet

“…Next, we give the extension of the tribonacci sequence to an infinite alphabet, which could be found in the much more general case in [12,13].…”

“…In this sense, these two sequences are similar. Hence, the infinite tribonacci sequence T may inherit some combinatorial properties of the tribonacci sequence t, which were studied sufficiently in [12,13]. Let L n := |φ n (0)| = |σ n (0)|, where L 0 = 1, L 1 = 2, and L 2 = 4.…”

“…Beyond this, extensions involving k letters gave birth to the k-bonacci sequence, a topic that has witnessed rigorous explorations [9][10][11]. Notably, recent contributions by Ghareghani, Mohammad-Noori, and Sharifani [12,13] presented a broadened scope by generalizing the k-bonacci sequence to an infinite alphabet. Their studies shed light on intriguing aspects such as palindrome complexity, square factors, and critical factors.…”

Mixing properties and entropy bounds of a family of Pisot random substitutions

“…The family of noble means substitutions consists of substitutions σ p : {a, b} Z → {a, b} Z induced by a substitution rule σ p : {a, b} * → {a, b} * , given by σ p (a) = a p b, and σ p (b) = a, where p ∈ N. The family of Pisa substitutions as defined by Baake and Grimm [4] is a set of substitutions of the form ς n : A Z → A Z , induced by a substitution rule ς n : A * → A * , where ς n (α i ) = α 1 α i+1 if i = n, and ς n (α n ) = α 1 , which are also called n-bonacci substitutions in the literature; compare [19,36]. Here, we consider the following generalisation which essentially combines the structures of these two families.…”

Some Properties of the Quasi-Tribonacci Sequence

“…Recently, Zhang, Wen and Wu [21] gave the extension of the Fibonacci sequence to the infinite alphabet N and studied its combinatorial properties, including the growth order, digit sum and several decompositions. Ghareghani, Mohammad-Noori and Sharifani [22,23] gave the generations of the m-bonacci sequence to the infinite alphabet and studied the palindrome complexity, square factors and critical factors. Given the extensive research and significant importance of the Fibonacci and Tribonacci sequences, here, we study the transformation of the Tribonacci morphism, which is called quasi-Tribonacci morphism.…”