A look and say sequence is an integer sequence in which each term after a chosen initial term is obtained by run-length
encoding, replacing each maximal run of identical digits
by its length followed by that digit. 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.
 | OEIS | 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, ... |
Numbering the initial term by 
, the numbers of digits in successive
terms of the sequence for 
are 1, 2, 2, 4, 6, 6, 8, 10, 14, 20, 26, 34, 46, 62, ...
(OEIS 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, ... (OEIS A022471). These sequences
are asymptotic to 
, where
 |  |  |
(1)
|
 |  |  |
(2)
|
 |  |  |
(3)
|
The quantity 
is known as Conway's constant (OEIS A014715),
and amazingly is given by the unique positive real root of the polynomial
 |
(4)
|
all of whose roots are illustrated above.
The same limiting growth ratio 
applies to every nonempty finite starting string
of digits except 22, a result which follows from the cosmological theorem (Ekhad and Zeilberger 1997).
Conway (1986) observed 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." Every finite string
of 1s, 2s, and 3s that does not contain four identical digits
in succession eventually "decays" into a compound of 92 special elements,
named after the chemical elements, as a consequence of the cosmological
theorem (Ekhad and Zeilberger 1997).
