I can't work out what this sequence is, though I am pretty sure I know
what sort of sequence it is. I would expect any of these:
A 0,1,0,2,0,1,1,3,0,3,1,1,2,4,0,5,1,2,2,2,3,5,0,6,1,5,2,3,3,1,4...
B 0,1,0,1,1,2,0,3,1,1,2,1,3,3,0,6,1,2,2,3,3,4,0,7,1,4,2,5,3,2,4...
C 0,1,0,1,1,2,0,1,2,3,1,3,0,2,3,2,2,5,1,4,0,1,5,1,4,3,3,5,2,6,1...
D 0,1,0,1,1,2,0,1,2,4,1,3,0,1,4,1,3,2,2,7,1,4,0,3,4,3,3,4,2,8,1...
The closest match, A, differs here:
A 0,1,0,2,0,1,1,3,0,3,1,1,2,4,0,5,1,2,2,2,3, 5,0,6,1,5,2,3,3,1,4...
0,1,0,2,0,1,1,3,0,3,1,1,2,4,0,5,1,2,2,2,3,1,4,1,5,5,0,8,1,5,2,3,3,2,4...
I suspect a typo but maybe someone has worked it out?
Spoiler: A is
http://oeis.org/A055186 and C is
http://oeis.org/A079686
the sequences B and D are related to A and C but the OP's sequence is
somewhere between A and B.