My process to solve this:
Letting n = the number of total cents, I will be solving for $a_n$.
So my goal is to find $a_8$, but before this I will have to look at the first three and try to find a pattern.
$a_1 = 1$: 1
$a_2 = 2$: 1+1 2
$a_3 = 4$: 1+1+1 1+2 2+1 3
$a_4$ is where it gets a little complicated so I counted by hand and ended up with 7: 1+1+1+1 1+2+1 1+1+2 1+3 2+1+1 2+2 3+1
This led to a pattern of $a_{n-1}$ of arrangements that start with 1, $a_{n-2}$ arrangements that start with 2 and finally $a_{n-3}$ arrangements that start with 3.
Using this same process for 5 and on, I got to $a_8$ and
$a_8 = a_7 + a_6 + a_5$ = 81.