On a family of automatic apwenian sequences

Abstract

An integer sequence ${\left\{a\right(n\left)\right\}}_{n\ge 0}$ is called apwenian if $a\left(0\right)=1$ and $a\left(n\right)\equiv a(2n+1)+a(2n+2)\left(\mathrm{mod}2\right)$ for all $n\ge 0$. The apwenian sequences are connected with the Hankel determinants, the continued fractions, the rational approximations and the measures of randomness for binary sequences. In this paper, we study the automatic apwenian sequences over different alphabets. On the alphabet $\{0,1\}$, we give an extension of the generalized Rueppel sequences and characterize all the 2-automatic apwenian sequences in this class. On the alphabet $\{0,1,2\}$, we prove that the only apwenian sequence, among all fixed points of substitutions of constant length, is the period-doubling like sequence. On the other alphabets, we give a description of the 2-automatic apwenian sequences in terms of 2-uniform morphisms. Moreover, we find two 3-automatic apwenian sequences on the alphabet $\{1,2,3\}$.