A Pattern That Never Ends, Yet is Always the Same... Eventually

John Conway, a freaking genius when it comes to numbers, tells the story of a time at a party when someone showed him this pattern:

1
11
21
1112
He says that he didn't figure out what the pattern was, so don't be ashamed if you didn't either. The pattern is, starting with 1, the amount of times that a number is shown in the line above. For example, the first line will be 1, then the next line is 11 because the number 1 appears one time. The line after that is 2 1 because there are 2 ones in the line above it, and so on. This is called a look and say pattern, because it requiers you to look at the pattern and say the numbers out loud to complete the pattern.

I thought, what if we were to extend this pattern? It would look like this.

1
11
21
1112
3112
211213
312213
212223
114213
31121314
41122314
31221324
21322314

What I discovered is that 13 lines is the most you can extend this pattern. After that, the pattern repeats itself infinitely or repeats the last few lines. The 13^th line has 2 1s, 3 2s, 2 3s and one 4- which makes the 14^th line 21322314, the same thing as the 13^th one.

This pattern also works if you start with another number. It even works if the 1^st line isn't a one digit number, like the pattern starting with 12:

12
1112
3112
211213
312113
311223
212223
114213
31121314
41122314
31221324
21322314
21322314
21322314

If you're really clever, you would have noticed that the last line is the same in both examples. Note that if a number is not in one line, it won't necessary be in the next one but once a number is introduced, it will be, at least onece, in every succeeding line.

I've tried doing starting with a number that has a zero in it and I got a pattern that eventually repeats the last two lines. I'm not going to show it here because they're too long. But you're welcome to try it yourself.

Anyways, that's what I've found so far. I can't find who told John Conway the original pattern, but here's the interview in which he introduces the pattern. Bye.