This is my favorite self-generating sequence:

1
11
21
1211
111221
312211
13112221
1113213211
31131211131221
13211311123113112211

[hint: just read each line out loud]

One of my favorites as well. It seems to pop up on rec.puzzles frequently as well.

-d

Hey, tie in with another thread ...
Level80 on notpron uses that exact sequence as a fake username/password clue.

TBTTFox

If I remember its called a "Morse-Thule sequence"

I've seen this called "Conway's Recursive Sequence" or similar, but not sure that he was the first to think of it. There are a number of variations on the idea.

You might be thinking of the Thue-Morse Sequence. There may well be a connection, though: each term of the CRS contains only 1,2,3 (so you don't get more than 3 consecutive digits that are the same), whereas the Thue-Morse sequence contains only 0 and 1, with never more than 2 consecutive digits (or even strings of digits) the same.

That's a new one for me. It's always fun looking these things up.

I found it called "Audioactive Sequence" and "Look and Say Sequence" in different places. Here's a link to MathWorld:

http://mathworld.wolfram.com/LookandSaySequence.html

Sandy