Short answer: If you use closed='left' and loffset="2T" then you’ll get what you expected:
series.resample('3T', label="left", closed='left', loffset="2T").sum()
2000-01-01 00:02:00 3
2000-01-01 00:05:00 12
2000-01-01 00:08:00 21
Long answer: (or why the results you got were correct, given the arguments you used) This may not be clear from the documentation, but open and closed in this setting is about strict vs non-strict inequality (e.g. < vs <=).
An example should make this clear. Using an interior interval from your example, this is the difference from changing the value of closed:
closed='right' => ( 3:00, 6:00 ] or 3:00 < x <= 6:00
closed='left' => [ 3:00, 6:00 ) or 3:00 <= x < 6:00
You can find an explanation of the interval notation (parentheses vs brackets) in many places like here, for example:
https://en.wikipedia.org/wiki/Interval_(mathematics)
The label parameter merely controls whether the left (3:00) or right (6:00) side is displayed, but doesn’t impact the results themselves.
Also note that you can change the starting point for the intervals with the loffset parameter (which should be entered as a time delta).
Back to the example, where we change only the labeling from ‘right’ to ‘left’:
series.resample('3T', label="right", closed='right').sum()
2000-01-01 00:00:00 0
2000-01-01 00:03:00 6
2000-01-01 00:06:00 15
2000-01-01 00:09:00 15
series.resample('3T', label="left", closed='right').sum()
1999-12-31 23:57:00 0
2000-01-01 00:00:00 6
2000-01-01 00:03:00 15
2000-01-01 00:06:00 15
As you can see, the results are the same, only the index label changes. Pandas only lets you display the right or left label, but if it showed both, then it would look like this (below I’m using standard index notation where ( on the left side means open and ] on the right side means closed):
( 1999-12-31 23:57:00, 2000-01-01 00:00:00 ] 0 # = 0
( 2000-01-01 00:00:00, 2000-01-01 00:03:00 ] 6 # = 1+2+3
( 2000-01-01 00:03:00, 2000-01-01 00:06:00 ] 15 # = 4+5+6
( 2000-01-01 00:06:00, 2000-01-01 00:09:00 ] 15 # = 7+8
Note that the first bin (23:57:00,00:00:00] is not empty, it’s just that it contains a single row and the value in that single row is zero. If you change ‘sum’ to ‘count’ this becomes more obvious:
series.resample('3T', label="left", closed='right').count()
1999-12-31 23:57:00 1
2000-01-01 00:00:00 3
2000-01-01 00:03:00 3
2000-01-01 00:06:00 2