Algebraic area of cubic lattice walks

We obtain an explicit formula to enumerate closed random walks on a cubic lattice with a specified length and algebraic area. The algebraic area of a closed cubic lattice walk is defined as the sum of the algebraic areas obtained from the walk's projection onto the three Cartesian planes. This enumeration formula can be mapped onto the cluster coefficients of three types of particles that obey quantum exclusion statistics with statistical parameters g=1, g=1, and g=2, respectively, subject to the constraint that the numbers of g=1 (fermions) exclusion particles of two types are equal.