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1113213211
The first row - 1 - contains one 1 -> 11
11 contains two 1's -> 21
21 contains one 2, one 1 ->1211
1211 contains one 1, one 2, two 1's ->111221
etcetera.
Building on that theory, it would go
111221 contains three 1's, two 2's, one 1-> 312211
312211 contains one 3,one 1,two 2's two 1's->13112221
13112221 contains one 1,one 3,three 2's, one 1,
therefore the next sequence of numbers would be ->1113213211
I will tell you what this is
A small history of this sequence
A005150 as a simple table
| n | a(n) | |
|---|---|---|
| 1 | 1 | |
| 2 | 11 | |
| 3 | 21 | |
| 4 | 1211 | |
| 5 | 111221 | |
| 6 | 312211 | |
| 7 | 13112221 | |
| 8 | 1113213211 | |
| 9 | 31131211131221 | |
| 10 | 13211311123113112211 | |
| 11 | 11131221133112132113212221 | |
| 12 | 3113112221232112111312211312113211 |
[1,11,21,1211,111221,312211,13112221,1113213211, 31131211131221,13211311123113112211, 11131221133112132113212221, 3113112221232112111312211312113211] |
Look and Say Sequence
The integer sequence beginning with a single digit in which the next term is obtained by describing the previous term. Starting with 1, the sequence would be defined by "1, one 1, two 1s, one 2 one 1," etc., and the result is 1, 11, 21, 1211, 111221, .... Similarly, starting the sequence instead with the digit 
for 
gives 
, 1
, 111
, 311
, 13211
, 111312211
, 31131122211
, 1321132132211
, ..., as summarized in the following table.
 | Sloane | sequence |
| 1 | A005150 | 1, 11, 21, 1211, 111221, 312211, 13112221, 1113213211, ... |
| 2 | A006751 | 2, 12, 1112, 3112, 132112, 1113122112, 311311222112, ... |
| 3 | A006715 | 3, 13, 1113, 3113, 132113, 1113122113, 311311222113, ... |
The number of digits in the 
th term of the sequence for 
are 1, 2, 2, 4, 6, 6, 8, 10, 14, 20, 26, 34, 46, 62, ... (Sloane's A005341). Similarly, the numbers of digits for the 
th term of the sequence for 
, 3, ..., are 1, 2, 4, 4, 6, 10, 12, 14, 22, 26, ... (Sloane's A022471). These sequences are asymptotic to 
, where
 |  |  | (1) |
 |  |  | (2) |
 |  |  | (3) |
The quantity 
is known as Conway's constant (Sloane's A014715), and amazingly is given by the unique positive real root of the polynomial
 | (4) |
all of whose roots are illustrated above.
In fact, the constant is even more general than this, applying to all starting sequences (i.e., even those starting with arbitrary starting digits), with the exception of 22, a result which follows from the cosmological theorem. Conway discovered that strings sometimes factor as a concatenation of two strings whose descendants never interfere with one another. A string with no nontrivial splittings is called an "element," and other strings are called "compounds." It is postulated that every string of 1s, 2s, and 3s that does not contain four of the same number in succession eventually "decays" into a compound of 92 special elements, named after the chemical elements.
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