#45

1
1 1
2 1
1 2 1 1
1 1 1 2 2 1

What's the next line?

Answer:

312211.

This is the "look and say" sequence in which each term after the first describes the previous term: one 1 (11); two 1s (21); one 2 and one 1 (1211); one 1, one 2, and two 1's (111221); and so on. See the look and say sequence http://mathworld.wolfram.com/LookandSaySequence.html entry on MathWorld for a complete write-up and the algebraic form of a fascinating related quantity known as Conway's constant http://mathworld.wolfram.com/ConwaysConstant.html