Abstract.
Let 
S(n) denote the total number of digits `1' in the binary expansions of the integers between 1 and 
n-1. The Trollope–Delange formula is a classical result which provides an explicit representation for 
S(n) in terms of the continuous, nowhere differentiable Takagi function. Recently, connections have been established between digital sums such as 
S(n) and certain functional equations associated with the Takagi function and its relatives. In the present paper we explore such a connection to derive a new, simple proof for the Trollope–Delange formula as well as for some of its generalizations involving power and exponential sums.
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