BASE SHIFTING SEQUENCES AND THE PALINDROME PROPERTY

Abstract For $M,N,m\in \mathbb {N}$ with $M\geq 2, N\geq 1$ and $m \geq 2$ , we define two families of sequences, $\mathcal {B}_{M,N}$ and $\mathcal {B}_{M,N,m}$ . The n th term of $\mathcal {B}_{M,N}$ is obtained by expressing n in base M and recombining those digits in base N . The n th term of $\mathcal {B}_{M,N,m}$ is defined by $\mathcal {B}_{M,N,m}(n):=\mathcal {B}_{M,N}(n) \; (\textrm {mod}\;m)$ . The special case $\mathcal {B}_{M,1,m}$ , where $N=1$ , yields the digit sum sequence $\mathbf {t}_{M,m}$ in base M mod m . We prove that $\mathcal {B}_{M,N}$ is the fixed point of a morphism $\mu _{M,N}$ at letter $0$ , similar to a property of $\mathbf {t}_{M,m}$ . Additionally, we show that $\mathcal {B}_{M,N,m}$ contains arbitrarily long palindromes if and only if $m=2$ , mirroring the behaviour of the digit sum sequence. When $m\geq M$ and N , m are coprime, we establish that $\mathcal {B}_{M,N,m}$ contains no overlaps.