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[[File:Conway's constant.svg|thumb|300px|The lines show the growth of the numbers of digits in the look-and-say sequences with starting points 23 (red), 1 (blue), 13 (violet), 312 (green). These lines (when represented in a [[logarithmic scale|logarithmic vertical scale]]) tend to straight lines whose slopes coincide with Conway's constant.]]
In [[mathematics]], the '''look-and-say sequence''' is the [[integer sequence|sequence of integers]] beginning as follows:

: 1, 11, 21, 1211, 111221, 312211, 13112221, 1113213211, ... {{OEIS|id=A005150}}.

To generate a member of the sequence from the previous member, read off the digits of the previous member, counting the number of digits in groups of the same digit.  For example:

* 1 is read off as "one 1" or 11.
* 11 is read off as "two 1s" or 21.
* 21 is read off as "one 2, then one 1" or 1211.
* 1211 is read off as "one 1, one 2, then two 1s" or 111221.
* 111221 is read off as "three 1s, two 2s, then one 1" or 312211.

The look-and-say sequence was introduced and analyzed by [[John Horton Conway|John Conway]].<ref>{{cite journal|last1=Conway|first1=John|title=The Weird and Wonderful Chemistry of Audioactive Decay|journal=Eureka|date=January 1986|volume=46|pages=5–16|url=http://www.archim.org.uk/archives/eureka/#46|url-status=dead|archiveurl=https://web.archive.org/web/20141011160655/http://www.archim.org.uk/archives/eureka/#46|archivedate=2014-10-11}}</ref>

The idea of the look-and-say sequence is similar to that of [[run-length encoding]].

If started with any digit ''d'' from 0 to 9 then ''d'' will remain indefinitely as the last digit of the sequence. For ''d'' different from 1, the sequence starts as follows:
: ''d'', 1''d'', 111''d'', 311''d'', 13211''d'', 111312211''d'', 31131122211''d'', …
Ilan Vardi has called this sequence, starting with ''d'' = 3, the '''Conway sequence''' {{OEIS|id=A006715}}. (for ''d'' = 2, see {{oeis|id=A006751}})<ref>[http://mathworld.wolfram.com/ConwaySequence.html Conway Sequence], [[MathWorld]], accessed on line February 4, 2011.</ref>

== Basic properties ==
[[File:Conway constant.png|frame|Roots of the Conway polynomial plotted in the [[complex plane]]. Conway's constant is marked with the [[Greek alphabet|Greek letter]] [[lambda]] ('''λ''').]]

=== Growth ===
The sequence grows indefinitely. In fact, any variant defined by starting with a different integer seed number will (eventually) also grow indefinitely, except for the [[degeneracy (mathematics)|degenerate]] sequence: 22, 22, 22, 22, … {{OEIS|id=A010861}}<ref name="Martin2006"/>

=== Digits presence limitation ===
No digits other than 1, 2, and 3 appear in the sequence, unless the seed number contains such a digit or a run of more than three of the same digit.<ref name="Martin2006">
{{cite journal
 |title=Look-and-Say Biochemistry: Exponential RNA and Multistranded DNA
 |first=Oscar
 |last=Martin
 |journal=American Mathematical Monthly
 |year=2006
 |volume=113
 |issue=4
 |pages=289–307	
 |publisher=Mathematical association of America
 |issn=0002-9890
 |url=http://www.uam.es/personal_pdi/ciencias/omartin/Biochem.PDF
 |archiveurl=https://web.archive.org/web/20061224154744/http://www.uam.es/personal_pdi/ciencias/omartin/Biochem.PDF
 |archivedate=2006-12-24
 |accessdate=January 6, 2010
 |doi=10.2307/27641915
}}</ref>

=== Cosmological decay ===
Conway's '''cosmological theorem''' asserts that every sequence eventually splits ("decays") into a sequence of "atomic elements", which are finite subsequences that never again interact with their neighbors.  There are 92 elements containing the digits 1, 2, and 3 only, which John Conway named after the [[chemical element]]s up to uranium, calling the sequence '''audioactive'''.  There are also two "[[transuranic]]" elements for each digit other than 1, 2, and 3.<ref name="Martin2006"/><ref>Ekhad, S. B., Zeilberger, D.: [http://www.ams.org/journals/era/1997-03-11/S1079-6762-97-00026-7/home.html Proof of Conway's lost cosmological theorem], Electronic Research Announcements of the American Mathematical Society, August 21, 1997, Vol. 5, pp.&nbsp;78–82. Retrieved July 4, 2011.</ref>

=== Growth in length ===
The terms eventually grow in length by about 30% per generation. In particular, if ''L''<sub>''n''</sub> denotes the number of digits of the ''n''-th member of the sequence, then the [[Limit (mathematics)|limit]] of the ratio <math>\frac{L_{n + 1}}{L_n}</math> exists and is given by
:: <math>\lim_{n \to \infty}\frac{L_{n+1}}{L_{n}} = \lambda</math>

where &lambda; = 1.303577269034... {{OEIS|id=A014715}} is an [[algebraic number]] of degree 71.<ref name="Martin2006"/> This fact was proven by Conway, and the constant &lambda; is known as Conway's [[Mathematical constant|constant]]. The same result also holds for every variant of the sequence starting with any seed other than 22.

==== Polynomial returning Conway's constant ====
Conway's constant is the unique positive [[real root]] of the following [[polynomial]]: {{OEIS|id=A137275}}

: <math>\begin{align}
&\,\,\,\,\,\,\,  x^{71}   &&  &&- x^{69}   &&- 2x^{68}  &&- x^{67}   &&+ 2x^{66}  &&+ 2x^{65}  &&+ x^{64}   &&- x^{63} \\
&- x^{62}  &&- x^{61}   &&- x^{60}   &&- x^{59}   &&+ 2x^{58}  &&+ 5x^{57}  &&+ 3x^{56}  &&- 2x^{55}  &&- 10x^{54} \\
&- 3x^{53} &&- 2x^{52}  &&+ 6x^{51}  &&+ 6x^{50}  &&+ x^{49}   &&+ 9x^{48}  &&- 3x^{47}  &&- 7x^{46}  &&- 8x^{45}  \\
&- 8x^{44} &&+ 10x^{43} &&+ 6x^{42}  &&+ 8x^{41}  &&- 5x^{40}  &&- 12x^{39} &&+ 7x^{38}  &&- 7x^{37}  &&+ 7x^{36}  \\
&+ x^{35}  &&- 3x^{34}  &&+ 10x^{33} &&+ x^{32}   &&- 6x^{31}  &&- 2x^{30}  &&- 10x^{29} &&- 3x^{28}  &&+ 2x^{27}  \\
&+ 9x^{26} &&- 3x^{25}  &&+ 14x^{24} &&- 8x^{23}  && &&- 7x^{21}  &&+ 9x^{20}  &&+ 3x^{19}  &&- 4x^{18}  \\
&- 10x^{17} &&- 7x^{16} &&+ 12x^{15} &&+ 7x^{14}  &&+ 2x^{13}  &&- 12x^{12} &&- 4x^{11}  &&- 2x^{10}  &&+ 5x^9     \\
& &&+ x^7      &&- 7x^6    &&+ 7x^5     &&- 4x^4     &&+ 12x^3    &&- 6x^2     &&+ 3x       &&- 6
\end{align}</math>

In his original article, Conway gives an incorrect value for this polynomial, writing − instead of + in front of <math>x^{35}</math>.<ref>Ilan Vardi, ''Computational Recreation in Mathematica''</ref> However, the value of {{mvar|λ}} given in his article is correct.

== Popularization ==
The look-and-say sequence is also popularly known as the '''Morris Number Sequence''', after cryptographer [[Robert Morris (cryptographer)|Robert Morris]], and the puzzle "What is the next number in the sequence 1, 11, 21, 1211, 111221?" is sometimes referred to as the '''Cuckoo's Egg''', from a description of Morris in [[Clifford Stoll]]'s book ''[[The Cuckoo's Egg]]''.<ref>[http://jamesthornton.com/fun/robert-morris-sequence.html Robert Morris Sequence<!-- Bot generated title -->]</ref><ref>[http://www.ocf.berkeley.edu/~stoll/number_sequence.html FAQ about Morris Number Sequence<!-- Bot generated title -->]</ref>

==Variations==
{{refimprove section|date=May 2012}}
There are many possible variations on the rule used to generate the look-and-say sequence.  For example, to form the "pea pattern" one reads the previous term and counts all instances of each digit, listed in order of their first appearance, not just those occurring in a consecutive block.  Thus, beginning with the seed 1, the pea pattern proceeds 1, 11 ("one 1"), 21 ("two 1s"), 1211 ("one 2 and one 1"), 3112 (three 1s and one 2), 132112 ("one 3, two 1s and one 2"), 311322 ("three 1s, one 3 and two 2s"), etc.  This version of the pea pattern eventually forms a cycle with the two terms 23322114 and 32232114.<ref>{{Cite web|url=http://codegolf.stackexchange.com/questions/4642/ascending-pea-pattern-generator|title=Ascending Pea Pattern generator|website=codegolf.stackexchange.com|access-date=2016-05-07}}</ref>

Other versions of the pea pattern are also possible; for example, instead of reading the digits as they first appear, one could read them in ascending order instead.  In this case, the term following 21 would be 1112 ("one 1, one 2") and the term following 3112 would be 211213 ("two 1s, one 2 and one 3").

These sequences differ in several notable ways from the look-and-say sequence.  Notably, unlike the Conway sequences, a given term of the pea pattern does not uniquely define the preceding term.  Moreover, for any seed the pea pattern produces terms of bounded length. This bound will not typically exceed 2 * [[radix]] + 2 digits and may only exceed 3 * [[radix]] digits in length for degenerate long initial seeds ("100 ones, etc"). For these maximum bounded cases, individual elements of the sequence take the form a0b1c2d3e4f5g6h7i8j9 for [[decimal]] where the letters here are placeholders for the digit counts from the preceding element of the sequence. Given that this sequence is infinite and the length is bounded, it must eventually repeat due to the [[pigeonhole principle]]. As a consequence, these sequences are always eventually [[periodic sequence|periodic]].

==See also==
* [[Gijswijt's sequence]]
* [[Kolakoski sequence]]
* [[Autogram]]

==References==
<references/>

== External links ==
* [https://www.youtube.com/watch?v=ea7lJkEhytA Conway speaking about this sequence] and telling that it took him some explanations to understand the sequence.
* [https://www.rosettacode.org/wiki/Look-and-say_sequence Implementations in many programming languages] on [[Rosetta Code]]
* {{MathWorld|urlname=LookandSaySequence|title=Look and Say Sequence}}
* [http://www.se16.info/js/looknsay.htm Look and Say sequence generator] p
* {{OEIS el|sequencenumber=A014715|name=Decimal expansion of Conway's constant}}
* [http://www.nathanieljohnston.com/2010/10/a-derivation-of-conways-degree-71-look-and-say-polynomial/ A Derivation of Conway’s Degree-71 “Look-and-Say” Polynomial]

{{Algebraic numbers}}

[[Category:Base-dependent integer sequences]]
[[Category:Algebraic numbers]]
[[Category:Mathematical constants]]
[[Category:John Horton Conway]]