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Apr 17, 2010 at 23:58 comment added Ross Tang @Douglas: I read your comment of random walk before giving the answer. I gave this answer, since unknown (google) asked for any approach for this problem. I am just giving a general approach for a complicated probability problem. Sorry that I can't solve the equations analytically. Would you mind giving me some guide in solving these kinds of partial difference equations? I don't know if generating function works in this case, due to the circle index.
Apr 9, 2010 at 18:43 comment added Douglas Zare You can make explicit formulas involving a single summation for your a(n,i). The number of walks which stay within an interval can be counted, and you want the walks from the origin to L or R which do stay inside the path excluding i, but which do not stay inside the path excluding i and the other neighbor. See mathoverflow.net/questions/16892/…
Apr 9, 2010 at 18:20 history answered Ross Tang CC BY-SA 2.5