3, 4, 7, 7, 13, 13, 21, 22, 31, 34, ?

The answer is $\boxed{\text{52}}$.

The sequence is generated by adding $1$ to the first number, then adding $4$ to the second number, then repeating the pattern. So the next number in the sequence is $34 + 4 = \boxed{\text{52}}$.

Option A, $42$, is incorrect because it is not the next number in the sequence.

Option B, $43$, is incorrect because it is not the next number in the sequence.

Option C, $51$, is incorrect because it is not the next number in the sequence.

Option D, $52$, is correct because it is the next number in the sequence.