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