How many ways are there to write the number 7 with the summands: 1, 2, and 3? For example, there are 7 ways to write the number 4:
{1 + 1 + 1 + 1} x 1
{2 + 1 + 1} x 3
{3 + 1} x 2
{2 + 2} x 1
I got the answer 44 by doing so:
{1 + 1 + 1 + 1 + 1 + 1 + 1} x 1
{2 + 1 + 1 + 1 + 1 + 1} x 6
{3 + 1 + 1 + 1 + 1} x 5
{2 + 2 + 1 + 1 + 1} x 12
{2 + 2 + 1 + 1} x 4
{3 + 2 + 2 + 1} x 12
{3 + 3 + 1} x 3
{3 + 2 + 2} x 3
The problem is, Im likely to miss a kind of summand, for example I forgot there is {2 + 2 + 1 + 1 + 1}, any tips on how to list them all? And how to get the cool number fonts people use here? Thanks!
In your counting the order of the summands is playing a rôle. Denote the number of representations of $n$ by summands $1$, $2$, $3$ by $a_n$. Any such representation has a last summand from $\{1,2,3\}$. Therefore we have the following recursion: $$a_n=a_{n-1}+a_{n-2}+a_{n-3}\qquad(n\geq3)\ .\tag{1}$$ The initial values are $a_0=a_1=1$, $a_2=2$, and we then obtain $a_3=4$, $a_4=7$. The general solution of $(1)$ is obtained through solution of its characteristic equation $\lambda^3-\lambda^2-\lambda-1=0$.