On the general solution of the Heideman-Hogan family of recurrences

We consider a family of nonlinear rational recurrences of odd order which was introduced by Heideman and Hogan. All of these recurrences have the Laurent property, implying that for a particular choice of initial data (all initial values set to 1) they generate an integer sequence. For these particular sequences, Heideman and Hogan gave a direct proof of integrality by showing that the terms of the sequence also satisfy a linear recurrence relation with constant coefficients. Here we present an analogous result for the general solution of each of these recurrences.