How many ways to make a score of N? In a game, there are three possible shots making points 3,5, and 10 respectively.
A player has scored exactly N points. What are the scoring sequences possible? This was the original problem I was given.
For instance, for N = 15, the four possible scoring sequences are:


*

*3,3,3,3,3

*5,5,5

*10,5

*5,10


This is a programming question and I could come up with a solution to figure out the exact sequences(using dynamic programming). The trouble I am having is with estimating the time complexity of this approach. More precisely, I am trying to figure out the number of solutions possible for a given N. For example, for N=15, the number of scoring sequences is 4. How can we mathematically come up with a bound for the number of scoring sequences of N if the three possible scores are 3, 5, and 10? 

You can define your function recursively:
$$f(n)=f(n-3)+f(n-5)+f(n-10),\\\text{for all }n>0$$
With initial conditions:
\begin{align}
f(n)&=0,&\text{for all }n<0\\
f(0)&=1
\end{align}
(There's only one sequence that gets us $0$ points: the empty sequence. Note that $n=0$ does not satisfy the recurrence relation.)
Proof for the recurrence relation: there are $f(n-3)$ sequences that begin with $3$, $f(n-5)$ sequences that begin with $5$, and $f(n-10)$ sequences that begin with $10$.
