[Author]
 Minabutdinov Aleksey R.

[Title]
 $q$-weighted analogue of the Trollope-Delange formula

[AMS Subj-class]
 11A63 Radix representation; digital problems
 39B22 Equations for real functions

[Keywords]
 $q$-weighted Trollope-Delange formula, Takagi-Landsberg curve

[Abstract]
 Let $s(n)$ denote the number of "$1$"s in the dyadic representation of a
 positive integer $n$ and sequence $S(n) = s(1)+s(2)+\dots+s(n-1)$.
 The Trollope-Delange formula is a classic result that represents the
 sequence $S$ in terms of the Takagi function. In this work, we replace
 the usual sum of digits $s(n)$ with its weighted analogue and obtain a
 variant of the Trollope-Delange formula for this case.

[Comments]
 Russian, 7 pp.
 Preprint is an updated variant of arXiv:1801.03120v3

[Contact e-mail]
 aminabutdinov@gmail.com