What’s the next number in this sequence?
1, 2, 2, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, …

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The next number is 9. This is Golomb’s sequence, a nondecreasing sequence of integers in which a_{n} is the number of times that n occurs in the sequence. We’re given a_{1} = 1 to start with, and this indicates that the number 1 appears only once in the whole sequence. That means that a_{2} is greater than 1. Setting a_{2} to 2 immediately tells us that 2 appears twice in the sequence, so a_{3} also must be 2. That tells us that 3 occurs twice in the sequence, and so on. In this way each number in the sequence is uniquely determined.
