.. and it is the least possible such number.

This defines the order.

This is not a 3 kyu kata for nothing. You may have to think hard.

A Golomb-type sequence describes itself .. every number in the .. sequence is the number of times the corresponding number from the given sequence appears

So, in 1 2 1 0 4 4 4 4 5 5 5 5 6 6 6 6 7 7 7 7, there is 1 zero, 2 ones, 1 two, 0 threes, 4 fours, 4 fives, 4 sixes, 4 sevens

The numbers I've written as digits come from the expected sequence itself; the numbers I've written as words come from the input sequence.

1 2 2 3 3 4 4 4 5 5 5 6 6 6 6 7 7 7 7 gives 0 zeroes, 1 one, 2 twos, 2 threes, 3 fours .., so it's not describing itself.