[alThor]

This problem (and your solution) remind me of the risk-of-ruin problems with integer payoffs. Such problems have been solved, e.g. when you lose one unit or win k units (2 in this case).

So one can also view the problem this way: consider the first time you "pass" any integer (say zero). There is a 50% chance that you hit it, and a 50% chance you jumped over it. If you jumped over it (and landed on one), you can ask the ROR question where you start with a bankroll of one unit, and are playing a game where you lose one or win two. Then,

P(hit zero) = .5 + .5*(ROR)

Solving the ROR uses the same style of difference equations you did, so this isn't really far from your answer; just making an observation on the connection.