In the study of the $2$-adic sum of digits function $S_2\left(n\right)$, the arithmetical function $u\left(0\right)=0$ and $u\left(n\right)={(-1)}^{n-1}$ for $n\ge 1$ plays a very important role. In this paper, we firstly generalize the relation between $S_2\left(n\right)$ and $u\left(n\right)$ to a bijective relation between arithmetical functions. And as an application, we investigate some aspects of the sum of digits functions $S_{𝒢}\left(n\right)$ induced by binary infinite Gray codes $𝒢$. We can show that the difference of the sum of digits function, $S_{𝒢}\left(n\right)-S_{𝒢}(n-1)$, is realized by an automaton. And the summation formula of the sum...

We analyse the roots of the polynomial $x^n-p{x}^{n-1}-qx-1$ for $p⩾q⩾1$. This is the characteristic polynomial of the recurrence relation $F_{k,p,q}\left(n\right)=p{F}_{k,p,q}(n-1)+q{F}_{k,p,q}(n-k+1)+F_{k,p,q}(n-k)$ for $n⩾k$, which includes the relations of several particular sequences recently defined. In the end, a matricial representation for such a recurrence relation is provided.