Sequence Puzzle Answer

1
11
21
1211
3112
132112
311322
232122
421311
14123113
41141223
24312213

What number comes next?

HINTS (read in order).

• Except for the first number, the rest of the digits are in pairs (this rule eventually breaks down, but much later).
• Each number can be generated from the previous one alone.
• Add up all the digits in the odd positions (1st, 3rd, 5th, etc) (this rule eventually breaks down, but much later).

ANSWER: 32142321.

RULE:
1. To generate the next sequence, you scan the previous sequence from left to right.
2. Each time you see a new digit, you write down the number of occurences of that digit in the string, followed by that digit. E.g., if you encounter '2' for the first time, and there are 17 occurences of '2' in the string, you write down "172".
3. Continue scanning the string looking for new digits.

E.g. begin with our last string "24312213" in the puzzle.
Initially, we encounter '2' and see that there are 3 occurences of '2'. So we write down "32".
Continuing, we encounter '4'. There is only 1 occurence of '4' and so we write down "14".
Next, we encounter '3', and writedown "23".
Next, we encounter '1', and writedown "21".
Next, we encounter '2', but this is not new and we write nothing. In fact, there are no more new digits.

BACKGROUND: It is implicit in the above description that each sequence is viewed as a sequence of digits, not as a sequence of integers.
We can also view each sequence as a sequence of positive integers. So the initial example ought to be written as "1", "1,1", "2,1", "1,2,1,1", etc. Furthermore, an alternative rule is to count the occurences of each number (not digit anymore) in increasing order. First count occurences of '1', then of '2', then of '3', etc. If a number does not occur, write down nothing. This is the version described in "On a Curious Proper of Counting Sequences" by V.Bronstein and A.Fraenkel, AMM vol.101,no.6(1994)560-563. They call such sequences "Counting Sequences" and prove that their version is ultimately periodic. The authors attribute the puzzle to John Conway.