2025/01/10 by Wenhui Ma, WANLI MA · 1 citation
Computer Science · Mathematics · #Computability, Logic, AI Algorithms #semigroups and automata theory #Mathematical Dynamics and Fractals
paper · doi:10.1017/s0004972724001187
Abstract For M,N,m∈ \mathbb N with M≥ 2, N≥ 1 and m ≥ 2 , we define two families of sequences, \mathcal BM,N and \mathcal BM,N,m . The n th term of \mathcal BM,N is obtained by expressing n in base M and recombining those digits in base N . The n th term of \mathcal BM,N,m is defined by \mathcal BM,N,m(n):=\mathcal BM,N(n) (\textrm mod m) . The special case \mathcal BM,1,m , where N=1 , yields the digit sum sequence \mathbf tM,m in base M mod m . We prove that \mathcal BM,N is the fixed point of a morphism μ M,N at letter 0 , similar to a property of \mathbf tM,m . Additionally, we show that \mathcal BM,N,m contains arbitrarily long palindromes if and only if m=2 , mirroring the behaviour of the digit sum sequence. When m≥ M and N , m are coprime, we establish that \mathcal BM,N,m contains no overlaps.