I got an idea but it's not complete yet. I will update it as soon as I find something else:
Also, I find something that applies to the first 4 number:
But after $48$ it doesn't work.
So after all I know one thing about the answer:
This is Very Tricky!
Here are the most obvious patterns that come to mind:
But then there's the metapattern, that is, the pattern of patterns, that doesn't look quite right:
That doesn't fit the puzzle's aesthetics at all, but what can we do?
So the initial values in the right branch were ...
Okay, so this puzzle has been bothering me for a few days. I still don't have a good solution, but I thought I'd share my thoughts about it.
I'm just hoping that somehow this helps someone see something, but really I have no idea and probably nothing here is particularly useful.
This was the only pattern I saw:
15, 18, 30, 39, 54, 69, 78, 90, 93 <-- starting
5 3 6 3 10 3 13 3 18 3 23 3 26 3 30 3 31 3 <-- factors (5*3 etc)
1 4 3 5 5 3 4 1 <-- difference between non-3
Now what jumped out at me was ...