  In the asymptotic analysis of regular sequences as defined by Allouche and
Shallit, it is usually advisable to study their summatory function because the
original sequence has a too fluctuating behaviour. It might be that the process
of taking the summatory function has to be repeated if the sequence is
fluctuating too much. In this paper we show that for all regular sequences
except for some degenerate cases, repeating this process finitely many times
leads to a ``nice'' asymptotic expansion containing periodic fluctuations whose
Fourier coefficients can be computed using the results on the asymptotics of
the summatory function of regular sequences by the first two authors of this
paper.
  In a recent paper, Hwang, Janson, and Tsai perform a thorough investigation
of divide-and-conquer recurrences. These can be seen as $2$-regular sequences.
By considering them as the summatory function of their forward difference, the
results on the asymptotics of the summatory function of regular sequences
become applicable. We thoroughly investigate the case of a polynomial toll
function.
