Given a grid graph G of size mn, we study the number i(m,n) of independent sets in G, as well as b(m,n), the number of maximal such sets. It turns out that the initial cases b(1,n) and b(2,n) lead to a Padovan and a Fibonacci sequence. To determine b(m,n) for m > 2 we present an adaptation of the transfer matrix method, well known for calculating i(m,n). Finally, we apply our method to obtain explicit values of b(m,n) for m=3,4,5 and provide the corresponding generating functions.
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