Answer: $10,15,9,20,12$


The possible scores that can be obtained are: $1,2,3,4,5,6,8,9,10,12,15,16,18,20,24,25,30,36$.

The third and fourth scores differ by $11$. The only pairs of numbers that do this are $(1,12)$, $(4,15)$, $(5,16)$, $(9,20)$ and $(25,36)$.

When these pairs are completed into sequences, they become:
$2,7,1,12,4$
$5,10,4,15,7$
$6,11,5,16,8$
$10,15,9,20,12$
$26,31,25,36,28$

Of these, only the fourth one uses only accessible numbers, so the sequence is $10,15,9,20,12$.