Editing Look-and-say sequence (section)

== 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, ... which remains the same size.<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
|jstor=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 92 naturally-occurring [[chemical element]]s up to [[uranium]], calling the sequence '''audioactive'''.  There are also two "[[transuranic]]" elements (Np and Pu) for each digit other than 1, 2, and 3.<ref name="Martin2006" /><ref>Ekhad, Shalosh B.; Zeilberger, Doron: [https://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> Below is a table of all such elements:
{| class="wikitable mw-collapsible mw-collapsed"
! colspan="5" |All "atomic elements" (Where E<sub>k</sub> is included within the derivate of E<sub>k+1</sub> except Np and Pu)<ref name="Conway-original-article" />
|-
!Atomic number (n)
!Element name (E<sub>k</sub>)
!Sequence
!Decays into<ref name="Martin2006" />
!Abundance
|-
|1
|H
|22
|H
|91790.383216
|-
|2
|He
|13112221133211322112211213322112
|Hf.Pa.H.Ca.Li
|3237.2968588
|-
|3
|Li
|312211322212221121123222112
|He
|4220.0665982
|-
|4
|Be
|111312211312113221133211322112211213322112
|Ge.Ca.Li
|2263.8860325
|-
|5
|B
|1321132122211322212221121123222112
|Be
|2951.1503716
|-
|6
|C
|3113112211322112211213322112
|B
|3847.0525419
|-
|7
|N
|111312212221121123222112
|C
|5014.9302464
|-
|8
|O
|132112211213322112
|N
|6537.3490750
|-
|9
|F
|31121123222112
|O
|8521.9396539
|-
|10
|Ne
|111213322112
|F
|11109.006696
|-
|11
|Na
|123222112
|Ne
|14481.448773
|-
|12
|Mg
|3113322112
|Pm.Na
|18850.441228
|-
|13
|Al
|1113222112
|Mg
|24573.006696
|-
|14
|Si
|1322112
|Al
|32032.812960
|-
|15
|P
|311311222112
|Ho.Si
|14895.886658
|-
|16
|S
|1113122112
|P
|19417.939250
|-
|17
|Cl
|132112
|S
|25312.784218
|-
|18
|Ar
|3112
|Cl
|32997.170122
|-
|19
|K
|1112
|Ar
|43014.360913
|-
|20
|Ca
|12
|K
|56072.543129
|-
|21
|Sc
|3113112221133112
|Ho.Pa.H.Ca.Co
|9302.0974443
|-
|22
|Ti
|11131221131112
|Sc
|12126.002783
|-
|23
|V
|13211312
|Ti
|15807.181592
|-
|24
|Cr
|31132
|V
|20605.882611
|-
|25
|Mn
|111311222112
|Cr.Si
|26861.360180
|-
|26
|Fe
|13122112
|Mn
|35015.858546
|-
|27
|Co
|32112
|Fe
|45645.877256
|-
|28
|Ni
|11133112
|Zn.Co
|13871.123200
|-
|29
|Cu
|131112
|Ni
|18082.082203
|-
|30
|Zn
|312
|Cu
|23571.391336
|-
|31
|Ga
|13221133122211332
|Eu.Ca.Ac.H.Ca.Zn
|1447.8905642
|-
|32
|Ge
|31131122211311122113222
|Ho.Ga
|1887.4372276
|-
|33
|As
|11131221131211322113322112
|Ge.Na
|27.246216076
|-
|34
|Se
|13211321222113222112
|As
|35.517547944
|-
|35
|Br
|3113112211322112
|Se
|46.299868152
|-
|36
|Kr
|11131221222112
|Br
|60.355455682
|-
|37
|Rb
|1321122112
|Kr
|78.678000089
|-
|38
|Sr
|3112112
|Rb
|102.56285249
|-
|39
|Y
|1112133
|Sr.U
|133.69860315
|-
|40
|Zr
|12322211331222113112211
|Y.H.Ca.Tc
|174.28645997
|-
|41
|Nb
|1113122113322113111221131221
|Er.Zr
|227.19586752
|-
|42
|Mo
|13211322211312113211
|Nb
|296.16736852
|-
|43
|Tc
|311322113212221
|Mo
|386.07704943
|-
|44
|Ru
|132211331222113112211
|Eu.Ca.Tc
|328.99480576
|-
|45
|Rh
|311311222113111221131221
|Ho.Ru
|428.87015041
|-
|46
|Pd
|111312211312113211
|Rh
|559.06537946
|-
|47
|Ag
|132113212221
|Pd
|728.78492056
|-
|48
|Cd
|3113112211
|Ag
|950.02745646
|-
|49
|In
|11131221
|Cd
|1238.4341972
|-
|50
|Sn
|13211
|In
|1614.3946687
|-
|51
|Sb
|3112221
|Pm.Sn
|2104.4881933
|-
|52
|Te
|1322113312211
|Eu.Ca.Sb
|2743.3629718
|-
|53
|I
|311311222113111221
|Ho.Te
|3576.1856107
|-
|54
|Xe
|11131221131211
|I
|4661.8342720
|-
|55
|Cs
|13211321
|Xe
|6077.0611889
|-
|56
|Ba
|311311
|Cs
|7921.9188284
|-
|57
|La
|11131
|Ba
|10326.833312
|-
|58
|Ce
|1321133112
|La.H.Ca.Co
|13461.825166
|-
|59
|Pr
|31131112
|Ce
|17548.529287
|-
|60
|Nd
|111312
|Pr
|22875.863883
|-
|61
|Pm
|132
|Nd
|29820.456167
|-
|62
|Sm
|311332
|Pm.Ca.Zn
|15408.115182
|-
|63
|Eu
|1113222
|Sm
|20085.668709
|-
|64
|Gd
|13221133112
|Eu.Ca.Co
|21662.972821
|-
|65
|Tb
|3113112221131112
|Ho.Gd
|28239.358949
|-
|66
|Dy
|111312211312
|Tb
|36812.186418
|-
|67
|Ho
|1321132
|Dy
|47987.529438
|-
|68
|Er
|311311222
|Ho.Pm
|1098.5955997
|-
|69
|Tm
|11131221133112
|Er.Ca.Co
|1204.9083841
|-
|70
|Yb
|1321131112
|Tm
|1570.6911808
|-
|71
|Lu
|311312
|Yb
|2047.5173200
|-
|72
|Hf
|11132
|Lu
|2669.0970363
|-
|73
|Ta
|13112221133211322112211213322113
|Hf.Pa.H.Ca.W
|242.07736666
|-
|74
|W
|312211322212221121123222113
|Ta
|315.56655252
|-
|75
|Re
|111312211312113221133211322112211213322113
|Ge.Ca.W
|169.28801808
|-
|76
|Os
|1321132122211322212221121123222113
|Re
|220.68001229
|-
|77
|Ir
|3113112211322112211213322113
|Os
|287.67344775
|-
|78
|Pt
|111312212221121123222113
|Ir
|375.00456738
|-
|79
|Au
|132112211213322113
|Pt
|488.84742982
|-
|80
|Hg
|31121123222113
|Au
|637.25039755
|-
|81
|Tl
|111213322113
|Hg
|830.70513293
|-
|82
|Pb
|123222113
|Tl
|1082.8883285
|-
|83
|Bi
|3113322113
|Pm.Pb
|1411.6286100
|-
|84
|Po
|1113222113
|Bi
|1840.1669683
|-
|85
|At
|1322113
|Po
|2398.7998311
|-
|86
|Rn
|311311222113
|Ho.At
|3127.0209328
|-
|87
|Fr
|1113122113
|Rn
|4076.3134078
|-
|88
|Ra
|132113
|Fr
|5313.7894999
|-
|89
|Ac
|3113
|Ra
|6926.9352045
|-
|90
|Th
|1113
|Ac
|7581.9047125
|-
|91
|Pa
|13
|Th
|9883.5986392
|-
|92
|U
|3
|Pa
|102.56285249
|-
! colspan="5" |Transuranic elements
|-
|93
|Np
|1311222113321132211221121332211n{{refn|group=note|name=first|n can be any digit 4 or above.}}
|Hf.Pa.H.Ca.Pu
|0
|-
|94
|Pu
|31221132221222112112322211n{{refn|group=note|name=first}}
|Np
|0
|}

=== 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 display="block">\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 constant'''. The same result also holds for every variant of the sequence starting with any seed other than 22.

==== Conway's constant as a polynomial root ====
Conway's constant is the unique positive [[real root]] of the following [[polynomial]] {{OEIS|id=A137275}}:
<math display="block">\begin{matrix}
             &           &\qquad            &           &\qquad            &           &\qquad            &            & +1x^{71}  &           \\
   -1x^{69}  & -2x^{68}  &   -1x^{67}  & +2x^{66}  &  +2x^{65}  &  +1x^{64} &  -1x^{63}  &  -1x^{62}  & -1x^{61}  & -1x^{60}  \\
   -1x^{59}  & +2x^{58}  &  +5x^{57}  & +3x^{56}  & -2x^{55}  & -10x^{54} &  -3x^{53}  &  -2x^{52}  & +6x^{51}  & +6x^{50}  \\
   +1x^{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}  & +1x^{35}  &  -3x^{34} & +10x^{33}  &  +1x^{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}   &           & +1x^{7}   &  -7x^{6}  &  +7x^{5}   &  -4x^{4}  & +12x^{3}   &  -6x^{2}   &  +3x^{1}  & -6x^{0}  \\
\end{matrix}
</math>

This polynomial was correctly given in Conway's original ''Eureka'' article,<ref name="Conway-original-article" />
but in the reprinted version in the book edited by Cover and Gopinath<ref name="Conway-original-article" /> the term <math>x^{35}</math> was incorrectly printed with a minus sign in front.<ref>
{{Cite book
  | last = Vardi
  | first = Ilan
  | title = Computational Recreations in Mathematica
  | publisher = [[Addison-Wesley]]
  | year = 1991
  | isbn = 0-201-52989-0
}}
</ref>