Recipe for solving linear discrete-time model for which N(t) is influenced by N(u) and N(v) u

Recipe for solving linear discrete-time model for which N(t) is influenced
by N(u) and N(v) u

For solving linear discrete time model of the form $$N_{t+1}=pN_{t}+c$$
one can first solve for the equilibrium: $$N_{equi}=\frac{c}{1-p}$$
Then define a variable d as the distance of the system from the
equilibrium: $$D_{t}=N_{t}-N_{equi}$$
The recursion equation for D is $D_{t+1}=pD_{t}$ because
$$D_{t+1}=n_{t+1}-N_{equi}=pN_{t}+c-N_{equi}$$
Replacing $N_{t}$ with $D_{t}-N_{equi}$ gives
$$D_{t+1}=pD_{t}+pN_{equi}+c-N_{equi}$$
Therefore, the general solution for D is $$D_{t}=p^tD_{0}$$
So, the general solution for Nis $$N_{t}=p^t(N_{0}-N_{equi})+N_{equi}$$
which also equals $$N_{t}=p^tc_{0}+(1-p^t)N_{equi}$$
Does it make sense?
What is the recipe for solving a model of the form:
$$N_{t}=pN_{t-1}+qN_{t-T}+c$$
T can take any integer values in the range [0;t]
The answer is maybe given here by Ron Gordon but I could not really
understand it.