  As we move towards future galaxy surveys, the three-point statistics will be
increasingly leveraged to enhance the constraining power of the data on
cosmological parameters. An essential part of the three-point function
estimation is performing triplet counts of synthetic data points in random
catalogues. Since triplet counting algorithms scale at best as
$\mathcal{O}(N^2\log N)$ with the number of particles and the random catalogues
are typically at least 50 times denser than the data; this tends to be by far
the most time-consuming part of the measurements. Here we present a simple
method of computing the necessary triplet counts involving uniform random
distributions through simple one-dimensional integrals. The method speeds up
the computation of the three-point function by orders of magnitude, eliminating
the need for random catalogues, with the simultaneous pair and triplet counting
of the data points alone being sufficient.
