A $q$-weighted analogue of the Trollope-Delange formula

Aleksei Minabutdinov

Let $s(n)$ denote the number of "$1$"s in the dyadic representation of a positive integer $n$ and sequence $S(n)=s(1)+s(2)+\cdots +s(n-1)$. The Trollope-Delange formula is a classi…