What is the next row of this pattern?

Question

I'm attempting to help my little sister with her Algebra 1 homework, and this problem showed up. I can't for the life of me figure out the pattern; I was hoping someone would be able to. Here's the given pattern:

1 1 1 2 1 1 1 1 2 3 1 1 2 2 1 1 2 1 3

Thanks!

Answer 1

This is a variation on the look and say sequence, it works as follows: Given the sequence $1231$ it has $2$ ones $1$ two and $1$ three. So the next line would be $211213$. The digits are said from smallest to largest (so we always start saying the number of appearances of digit 1)

Answer 2

As Modded Bear points out, this is a modified version of the classical look and say sequence (CLASS). However, it is not clear which modification is intended.

CLASS requires that digits are placed in the order they appear, repeated blocks are counted separately. So it begins

1 11 21 1211

(which is different from yours in the fourth line: $1112$). From here, the two blocks of ones are counted at different times; the next line is not $3112$ but is

111221

It is possible that your sequence attempts to "split up blocks", but not display them in the order they appear, but instead in numerical order based on the number they are counting. This would explain why the fourth line is $1211$, and it would be consistent with the remaining lines. There are still several choices that would need to be made; one way to extend the sequence is as follows:

1 11 21 11 12 31 12 21 12 13 11 21 22 13 11 11 21 12 22 13 11 21 41 12 32 13 ...

(You may find it interesting to decipher the choices I made and investigate other possible choices)

However, what I suspect is happening is that order is that the block effect is being ignored completely, in which case the sequence proceeds:

1 11 21 11 12 31 12 21 12 13 31 22 13 21 22 23 11 42 13 31 12 13 14 ...

Note that both of these are qualitatively different than CLASS: it is an easy exercise to prove that CLASS will never contain a $4$. Intuitively, there is no bound on the size of numbers in the first one, but I'm not sure how one would prove this.

Actually, the second one is extremely different; it is finite. Continuing from where we left off, we only need to go a few more steps before we find a "self-descriptive" number.

41 12 23 14 21 22 13 24 21 42 13 14 31 22 13 24 21 32 23 14 21 32 23 14 21 32 23 14 ...

Cheers!