Find the rest of this brain teaser sequence

[Bartholomew]

Fill in the question marks.

0   1   1   1   1   1   1   1   0
0   1   2   2   2   2   2   1   0
0   1   3   4   4   4   3   1   0
0   1   4   6   7   6   4   1   0
0   1   5   8   9   8   5   1   0
0   1   6   9   10  9   6   1   0
?   ?   ?   ?   ?   ?   ?   ?   ?
?   ?   ?   ?   ?   ?   ?   ?   ?

edit: I deleted the first row of zeroes.

 

[Alkatran]

rates of change:
n   n   n   n   n   n   n   n   n
0   0   1   1   1   1   1   0   0
0   0   1   1   1   1   1   0   0
0   0   1   2   2   2   1   0   0
0   0   1   2   3   2   1   0   0
0   0   1   2   2   2   1   0   0
0   0   1   1   1   1   1   0   0
0   0   1   1   1   1   1   0   0
0   0   1   1   1   1   1   0   0
0   0   0   0   0   0   0   0   0
values:
0   0   0   0   0   0   0   0   0
0   1   1   1   1   1   1   1   0
0   1   2   2   2   2   2   1   0
0   1   3   4   4   4   3   1   0
0   1   4   6   7   6   4   1   0
0   1   5   8   9   8   5   1   0
0   1   6   9   10  9   6   1   0
0   1   7  10  11  10  7   1   0
0   1   8  11  12  11  8   1   0

 

[Bartholomew]

You got annoyingly close with such a simple method , but you didn't quite get it. This isn't as good a puzzle as I thought it might be. You might get the second-to-last row (you didn't this time), but you'll never get the last row with the method you're using. Good try, though.

 

[Bartholomew]

I'll just give the answer for the last row: it's all zeros. You could probably get the second-to-last row in 1 guess just by the method you've used. Meanwhile I'll try to think of a harder sequence having to do with this kind of thing.

 

[Alkatran]

The problem is that it's an arbitrary pattern. With such a small sequence (or any sequence, really) of numbers you can come up with infinite patterns to match it. You can't say an answer is 'wrong', because it matches at least one of those patterns! I could have typed in random digits if I wanted to.

Good pattern problems look complicated but end up having very simple solutions.

 

[Bartholomew]

Well, it is generated by a fairly simple method. The problem is that it can be solved mechanically; the generating method is not chaotic enough to rule out simple ways of solving it.