  A highly accurate and efficient method to compute the expected values of the
count, sum, and squared norm of the sum of the centre vectors of a random
maximal sized collection of non-overlapping unit diameter disks touching a
fixed unit-diameter disk is presented. This extends earlier work on R\'enyi's
parking problem [Magyar Tud. Akad. Mat. Kutat\'{o} Int. K\"{o}zl. 3 (1-2),
1958, pp. 109-127]. Underlying the method is a splitting of the the problem
conditional on the value of the first disk. This splitting is proven and then
used to derive integral equations for the expectations. These equations take a
lower block triangular form. They are solved using substitution and
approximation of the integrals to very high accuracy using a polynomial
approximation within the blocks.
