Given $n \in \mathbb{N}$ how many ways can one write $n=a+2b+3c$ for $a,b,c \in \mathbb{N}$.

I have an idea as if I use a a 3-tuple to represent $(a,b,c)$, I can list all of them using two functions $f((a,b,c)) = (a-2,b+1,c)$ and $g((a,b,c)) = (a-1,b-1,c+1)$. Which makes a nice pattern that should be easy to compute for any n however I'm struggling finding an actual equation that works everytime. Right now I have one that works sometimes being; Let $x = \left \lfloor{n/2}\right \rfloor $, $y = \left \lfloor{n/6}\right \rfloor $ and $z = x-y$. Then the function $f(n)$ is the count.
$f(n) = 1+x+(c/2)(c+1)+bc-b(b+1)$. However I know it's wrong.

EDIT:
Sorry forgot to add that f(n) should be a function of just n.