Orange Juice, Milk, or Beer

Question

Suppose that we have $n$ dollars and that each day we buy either orange juice for a dollar, milk for two dollars, or beer for two dollars. If $R_n$ is the number of ways of spending all the money, show that $$R_n=R_{n-1}+2R_{n-2}.$$

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How can I see this in another way? I mean like in a steps-along-a-grid kind of way, or maybe that's useless...

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Order is taken into account, as, for example there are $11$ ways to spend $ \$4$: $$MB, BM, OOM, OOB, OMO, OBO, MOO, BOO, OOOO, MM, BB$$

Answer 1

Hint:

If order is taken into account, then notice that to use $n$ dollars, you will make your final purchase with either $1$ dollar remaining, or $2$ dollars remaining. In the first case, you can only purchase orange juice, and in the second, you have two options, milk or beer.

See if you can translate the previous sentences into a recurrence relation.