Problem 5. A vending machine dispensing books of stamps accepts only $1 coins, $1 bills, and $2 bills. Let an denote the number of ways of depositing n dollars in the vending machine, where the order in which the coins and bills are deposited matters. (a) Find a recurrence relation for an and give the necessary initial condition(s). (b) Find an explicit formula for an by solving the recurrence relation in part (a).

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