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The series $a_n$ is defined by a recursive formula $a_n = a_{n-1} + a_{n-3}$ and its base case is $a_1 = 1 \ a_2 = 2 \ a_3 = 3$.

Prove that every natural number can be written as a sum (of one or more) of different elements of the series $a_n$.

Now, I know that is correct intuitively but I don't know how to prove that.

Generally, I have some problem of understanding the concept of recursion.

Thanks.