Complete the pattern 4,15,7,12,10,9 ? ? ? show me how you got your work

Given the pattern:
4,15,7,12,10,9,..
Every other term after the first term is increasing by 3. Thus continuing the pattern we shall have:
4, 15, 7, 12, 10, 9, 13, 11,16
thus the missing numbers will be:
13,11,16

Split this pattern into two sub-patterns.

We can see that odd position terms first, third, and fifth terms are increasing by 3.

We also can see that even position terms second, fourth, and sixth terms are decreasing by 3.

So the remaining term is seventh, eighth, and ninth terms. So from pattern above, seventh and ninth is continue after odd position term, so that would be 13 then 16.

eighth term is continue after even position term. That would be 6.

So then the remaining pattern would be 13, 6, 16.