I am looking for information on counting the possible ways of summing numbers. For example, suppose you can use 1,2, and 4. How many possible combinations are there to create any number?
For instance,
For 1: $ 1$
For 2: $ 2, 1+1$
For 3: $2+1, 1+1+1$
For 4: $4, 2+2, 1+1+2$
For 5: $4+1, 2+2+1, 2+1+1+1, 1+1+1+1+1$
This is for a question I am working on a for a first year linear algebra class. We have just been introduced to writing proofs and logical reasoning etc. The numbers we are actually given are 1,2, and 3. However, I am trying to get background information first so I can attempt to solve it myself. I have taken some statistics so I am aware of combinations, but not in this form.
I realize that for every number $n$, there is a combination $\sum_{1}^{n}1$ that equals the number. I made some tables of different combinations for numbers 1-9. I am trying different ways of organizing the different combinations. Any help would be appreciated.