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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 nth term of 
$\mathcal {B}_{M,N}$ is obtained by expressing n in base M and recombining those digits in base N. The nth term of 
$\mathcal {B}_{M,N,m}$ 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.
