I currently have an open question about counting the possible ways of summing numbers. I am still exploring all the ideas provided - those within my level of understanding. This is a question involving looking into one avenue and possible solution. The question involves finding the possible ways of creating a number $n\in \mathbb{I}$ only by adding 1, 2, or 3. For example, for the number 4:
$1+1+1+1, 1+1+2, 2+2, 3+1$
There are four possible combinations. My question involves breaking down the number $n$ using as many 3's as possible. For instance, the number 12 can be written $3+3+3+3.$ Any number greater than or equal to 3 can be broken into
$3+3+\cdot \cdot \cdot+3$ or $3+3+\cdot \cdot \cdot+3+2$ or $3+3+\cdot \cdot \cdot+3+1$
For the first case, it would be a number $n$ that when divided by three yields an integer. Suppose $n=12$, then $n$ can be written as $3+3+3+3$. However, each of these 3's can be written as $3$, $2+1$, or $1+1+1$. That is, there are three possible ways of summing to 3.
How can I calculate the possible combinations? I've thought for the case that $n=12$, there are four 3's that can be either $1+1$, or $1+2$, or $1+1+1$. Can any given number be broken down into a number of 3's, then the possible combinations of $3, 2+1, 1+1+1$ calculated? I would also have to take into account the cases where there would be a remainder of 1 or 2 after dividing by 3.
Does this make any sense?