  A generic uniformly distributed random sequence on the unit interval has
Poissonian pair correlations. At the same time, there are only very few
explicitly known examples of sequences with this property. Moreover, many types
of deterministic sequences, which are important in other contexts of
equidistribution theory, have been proven to fail having the Poissonian pair
correlation property. In all known examples for the non-Poissonian case, rather
sophisticated arguments were used to derive information on the limiting pair
correlation function. In this paper, we derive therefore the first elementary
such example, namely for the sequence $x_n := \left\{
\frac{\log(2n-1)}{\log(2)} \right\}$, which is also a low-dispersion sequence.
The proof only heavily relies on a full understanding of the gap structure of
$(x_n)_{n \in \mathbb{N}}$. Furthermore, we discuss differences to the weak
pair correlation function.
