How many ways can you add the numbers 1, 2, and 3 to create a number n?

Question

I would like to find an equation in which, given a number n, you can find the amount of ways to add 1, 2, and 3 to create the number. The commutative rule doesn't apply; for example, 1 + 1 + 3 is a different arrangement than 1 + 3 + 1. Let's take the number 4 for an example. There are 7 ways to combine the numbers 1, 2, and 3 to create 4: 1 1 1 1 1 1 2 1 2 1 2 1 1 1 3 3 1 2 2

Any ideas?

Answer 1

Let $a_n$ be the number of such partitions. Then $a_1 = 1, a_2 = 2, a_3 = 4$, and by conditioning on the size of the first part, we find that $$a_n = a_{n-1} + a_{n-2} + a_{n-3}.$$ Do you know how to solve this recurrence relation?