29.
Basically each time you add another line, you add the previous amount of lines placed, but we get to start at 1.
0 lines = 1 section
1 line = 0 lines + 1 section = 2 sections
2 lines = 1 line + 3 sections = 4 sections... etc
3 lines = 4 +3 = 7 sections
4 lines = 7 + 4 = 11 sections
5 lines = 11 + 5 = 16 sections
6 lines = 16 + 6 = 22 sections
7 lines = 22 + 7 = 29 sections
The pattern for this one also matches how you would do the 13th. 1, 2, 4, 7, 11, 16, 22, 29, 37...
where f(x) is the number of sections with x lines, f(x) = (x*(x+1) +2)/2