  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.
