Number sequence (increasing integers with special cases): determine the next term. Sequence given: 2, 1, 2, 4, 4, 5, 6, 7, 8, 8, 10, 11, ? Infer the inclusion/omission rule and compute the next number. Choose the correct value.

Observation

• Natural numbers appear in increasing order but odd multiples of 3 (3, 9, 15, …) are omitted.
• Powers of 2 (2, 4, 8, …) appear twice.

Check the run
… 7, 8, 8, 10, 11, (skip 9 as it is an odd multiple of 3) → next allowed integer is 12.

Note
This explains earlier duplications at 2, 4 and 8, and the absence of 3 and 9.