Initial condition for recurrence relation

Question

I have a question regarding the solution of this problem.

The problem is:

Suppose that we have n dollars to use to buy either orange juice for 1, milk for 2, or beer for 2, and the order in which we make the purchases matter. Let $R_n$ be the number of ways to spend all the money. Derive a recurrence relation and initial conditions for the sequence $R_0$, $R_1$, $R_2$, ... .

The solution is: $R_n$ = $R_{n-1}$ + 2$R_{n-2}$, with $R_0$ = 1 and $R_1$ = 1.

Now I understand how to get the recurrence relation, and I understand the initial condition for $R_1$=1, but I don't understand why $R_0$ = 1 as well. If you have n = 0, which represents 0 dollars, shouldn't you have 0 ways to spend it?

Thanks!

Answer 1

We clearly have $\;R_1=1\;,\;\;R_2=3\;$ , so we get $\;R_0\;$ from the recursive relation:

$$3=R_2=R_1+2R_0=1+2R_0\implies R_0=1$$

Many times $\;R_0\;$ doesn't have an intuitive meaning connected with the problem, is just something helpful to make work the recursion