Insert the missing number. 16, 33, 65, 131, 261, (....)

This is a mixed multiplicative-additive number series: each term is generated from the one before it by doubling it and then adding or subtracting 1, with the sign of the adjustment flipping at every step.

Applying this rule term by term:

1. 16 × 2 + 1 = 33

2. 33 × 2 − 1 = 65

3. 65 × 2 + 1 = 131

4. 131 × 2 − 1 = 261

5. 261 × 2 + 1 = 523

Independent cross-check: pairing terms two positions apart, 33 and 131 satisfy (term × 4) − 1 = next such term (33 × 4 − 1 = 131). Applying the same relationship to the next pair, 131 × 4 − 1 = 523, which matches the step-by-step result above — confirming the missing term is 523.