I'd like to find a recurrence relation/recursive definition that can count the total number of combinations s.t. $3x+4y+5z=a$
In other words, given a value $a$, how many sets of $x,y,z \in \mathbb{N}$ exist for such an $a$ so that the equation is true. I should mention that $\mathbb{N}$ includes the number $0$.
What I thought of doing was having a function like (I wasn't sure how to make a piecewise function):
$f(a)=0$, when $a<0$
$f(a)=1$, when $a=1$
$f(a)=f(a-3)+f(a-4)+f(a-5)$
But obviously, a counterexample pops up rather quickly so that isn't the answer.
I'd like to find a function that only uses a single parameter but is a function like this even possible?