  The goal of this article is to study the discrepancy of the distribution of
arithmetic sequences in arithmetic progressions. We will fix a sequence
$\A=\{\a(n)\}_{n\geq 1}$ of non-negative real numbers in a certain class of
arithmetic sequences. For a fixed integer $a\neq 0$, we will be interested in
the behaviour of $\A$ over the arithmetic progressions $a \bmod q$, on average
over $q$. Our main result is that for certain sequences of arithmetic interest,
the value of $a$ has a significant influence on this distribution, even after
removing the first term of the progressions.
