Following the suggestion by @acer, I'm reposting my new question as a reply on this original question.
Now, we want the (1, 1)th entry of the mth power of the matrix G_(0, n). When I executed the procedure given by @Carl Love, I observe that the program keeps running even for moderately small vales of m and n, say m=6 and n=9 without any response.
One more thing that concerns me, is that I observe that G_(0, n) is quite sparse, and I feel that @Carl Love's procedure calculates the mth power of G_(0, n) and then only takes the (1, 1)th entry. Please correct me if I'm wrong.
I was thinking that suppose we computed the (m-1)th power of G_(0, n) and denoted it by M. Then take the first column of M and the first row of G_(0, n) and multiply them, we would get only the (1, 1)th entry of the mth power of G_(0, n).
But, I'm unsure whether this is more efficient than the usual method. If yes, I'm stuck on how to code this. Any help would be appreciated.