Recurrence relation for feasible column-vector counts

Derive a recurrence relation for the number of feasible labeled column vectors of length m, denoted by |T(m)|, in the labeling framework for computing the integer {2}-domination number of grid graphs G_{m,n}.

Background

The algorithm for computing the integer {2}-domination number of a grid graph G_{m,n} uses feasible column vectors whose entries encode the local status of vertices under a function taking values in {0,1,2}. The paper lists the initial values |T(1)|=6, |T(2)|=23, |T(3)|=95, |T(4)|=389, |T(5)|=1595, |T(6)|=6538, and |T(7)|=26802, but does not provide a recurrence generating these quantities. Establishing such a recurrence would clarify the combinatorial structure and growth of the state space underlying the proposed algorithm.