Ways to get a sum of $59$ from $1$, $4$ and $6$ How many ways can you get a sum of $59$ using only the numbers $1$, $4$ and $6$?
Note: In this case, $1 + 4$ and $4 + 1$ are two different ways to sum $5$.
For example, there are $5$ ways to get $6$:
$1+1+1+1+1+1$
$1+1+4$
$4+1+1$
$1+4+1$
$6$
You can set up a recurrence: the last term of the sum is either $1$ or $4$ or $6$, so after the first few initial terms we have (using $c(n)$ for the count of the number of ways of getting the sum $n$) $$c(n)=c(n-1)+c(n-4)+c(n-6)$$This can be solved in various standard ways, or simply computed out.
You can find it from a spreadsheet.  Using fill down, list the numbers from -6 through 59 n a column.  The next column will have the number of ways to express the number in the first column.  Put $1$ opposite $0$ as there is one way to express $0$.  Then in the line below put =Up1+Up4+Up6 to get the number of ways to express $1$, which is $1$.  We left the blank lines so this works even for $1$.  Copy down and you have your answer of $2890628972$