Let $A_n$ be the set of all finite words over the alphabet $\{1,2,3\}$ whose digits sum to $n$. I am trying to find a recurrence relation with initial conditions for $|A_n|$. I have checked the first few $n$ and got that $|A_1|=1$ , $|A_2|=2$, $|A_3|=4$, $|A_4|=7$, $|A_5|=13$, $|A_6|=24$. I can't seem to find a pattern that allows me to create a recurrence relation, nor I am I sure how I could check that such a pattern would be correct.
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