Kamiya, Murata: Certain codes related to generalized paperfolding sequences

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Yuichi Kamiya; Leo Murata
Certain codes related to generalized paperfolding sequences
Journal de théorie des nombres de Bordeaux, 27 no. 1 (2015), p. 149-169, doi: 10.5802/jtnb.896
Article PDF | Reviews MR 3346967
Class. Math.: 11B85, 11A25
Keywords: Paperfolding sequence, Numeration system, Sum of digits function

Résumé - Abstract

Let RBC be the reflected binary code, which is also called the Gray code, $S_{\rm RBC}$ be the sum of digits function for RBC, and $\lbrace P_{{\bf b}_{0}}(n)\rbrace _{n=1}^{\infty }$ be the regular paperfolding sequence. In their previous work the authors proved that the difference function of the sum of digits function for RBC, $\lbrace S_{\rm RBC}(n)-S_{\rm RBC}(n-1)\rbrace _{n=1}^{\infty }$, coincides with $\lbrace P_{{\bf b}_{0}}(n)\rbrace _{n=1}^{\infty }$. From an infinite sequence ${\bf b}=\lbrace b_{k}\rbrace _{k=0}^{\infty }$ with $b_{k}\in \lbrace -1,1\rbrace $, one can construct an infinite sequence $\lbrace P_{\bf b}(n)\rbrace _{n=1}^{\infty }$ which is called the generalized paperfolding sequence with respect to ${\bf b}$. In this paper, when we assume ${\bf b}$ is periodic, we propose a new numeration code ${\mathcal{C}}_{\bf b}$, and study some properties of the code ${\mathcal{C}}_{\bf b}$ in Theorem 1.2. We can prove that the difference function of the sum of digits function $S_{{\mathcal{C}}_{\bf b}}$ for ${\mathcal{C}}_{\bf b}$, $\lbrace S_{{\mathcal{C}}_{\bf b}}(n)-S_{{\mathcal{C}}_{\bf b}}(n-1)\rbrace _{n=1}^{\infty }$, coincides with the generalized paperfolding sequence $\lbrace P_{\bf b}(n)\rbrace _{n=1}^{\infty }$ (Theorem 1.1). We also give an exact formula for the average of $S_{{\mathcal{C}}_{\bf b}}$ in Theorem 1.3.

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