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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) tend to straight lines whose slopes coincide with Conway's constant.]]
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| In [[mathematics]], the '''look-and-say sequence''' is the [[integer sequence|sequence of integers]] beginning as follows: | |
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| : 1, 11, 21, 1211, 111221, 312211, 13112221, 1113213211, ... {{OEIS|id=A005150}}.
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| 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:
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| * 1 is read off as "one 1" or 11.
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| * 11 is read off as "two 1s" or 21.
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| * 21 is read off as "one 2, then one 1" or 1211.
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| * 1211 is read off as "one 1, then one 2, then two 1s" or 111221.
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| * 111221 is read off as "three 1s, then two 2s, then one 1" or 312211.
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| The look-and-say sequence was introduced and analyzed by [[John Horton Conway|John Conway]] in his paper "The Weird and Wonderful Chemistry of Audioactive Decay" published in ''[[Eureka (University of Cambridge magazine)|Eureka]]'' 46, 5–18 in 1986.
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| The idea of the look-and-say sequence is similar to that of [[run-length encoding]].
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| If we start 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:
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| : ''d'', 1''d'', 111''d'', 311''d'', 13211''d'', 111312211''d'', 31131122211''d'', …
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| Ilan Vardi has called this sequence, starting with ''d'' = 3, the '''Conway sequence''' {{OEIS|id=A006715}}.<ref>[http://mathworld.wolfram.com/ConwaySequence.html Conway Sequence], [[MathWorld]], accessed on line February 4, 2011.</ref>
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| == Basic properties ==
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| [[File:Conway constant.png|frame|Roots of the Conway polynomial plotted in the [[complex plane]].]]
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| * 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, ….<ref name="Martin2006"/>
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| * 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">
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| {{cite news
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| |title=Look-and-Say Biochemistry: Exponential RNA and Multistranded DNA
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| |chapter=Introduction
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| |first=Oscar
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| |last=Martin
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| |journal=American mathematical monthly
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| |year=2006
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| |volume=113
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| |issue=4
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| |pages=289–307
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| |publisher=Mathematical association of America
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| |issn=0002-9890
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| |url=http://www.uam.es/personal_pdi/ciencias/omartin/Biochem.PDF
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| |archiveurl=http://web.archive.org/web/20061224154744/http://www.uam.es/personal_pdi/ciencias/omartin/Biochem.PDF
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| |archivedate=2006-12-24
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| |accessdate=January 6, 2010
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| }}</ref>
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| * Conway's '''cosmological theorem''': Every sequence eventually splits 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 natural [[chemical element]]s. 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. 78-82. Retrieved July 4, 2011.</ref>
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| * 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
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| :: <math>\lim_{n \to \infty}\frac{L_{n+1}}{L_{n}} = \lambda</math>
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| : where λ = 1.303577269... is an [[algebraic number]] of degree 71.<ref name="Martin2006"/> This fact was proven by Conway, and the constant λ 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.
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| Conway's constant is the unique positive real root of the following [[polynomial]]:
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| : <math>\begin{align}
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| &\,\,\,\,\,\,\, x^{71} && &&- x^{69} &&- 2x^{68} &&- x^{67} &&+ 2x^{66} &&+ 2x^{65} &&+ x^{64} &&- x^{63} \\
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| &- x^{62} &&- x^{61} &&- x^{60} &&- x^{59} &&+ 2x^{58} &&+ 5x^{57} &&+ 3x^{56} &&- 2x^{55} &&- 10x^{54} \\
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| &- 3x^{53} &&- 2x^{52} &&+ 6x^{51} &&+ 6x^{50} &&+ x^{49} &&+ 9x^{48} &&- 3x^{47} &&- 7x^{46} &&- 8x^{45} \\
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| &- 8x^{44} &&+ 10x^{43} &&+ 6x^{42} &&+ 8x^{41} &&- 5x^{40} &&- 12x^{39} &&+ 7x^{38} &&- 7x^{37} &&+ 7x^{36} \\
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| &+ x^{35} &&- 3x^{34} &&+ 10x^{33} &&+ x^{32} &&- 6x^{31} &&- 2x^{30} &&- 10x^{29} &&- 3x^{28} &&+ 2x^{27} \\
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| &+ 9x^{26} &&- 3x^{25} &&+ 14x^{24} &&- 8x^{23} && &&- 7x^{21} &&+ 9x^{20} &&+ 3x^{19} &&- 4x^{18} \\
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| &- 10x^{17} &&- 7x^{16} &&+ 12x^{15} &&+ 7x^{14} &&+ 2x^{13} &&- 12x^{12} &&- 4x^{11} &&- 2x^{10} &&+ 5x^9 \\
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| & &&+ x^7 &&- 7x^6 &&+ 7x^5 &&- 4x^4 &&+ 12x^3 &&- 6x^2 &&+ 3x &&- 6
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| \end{align}</math>
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| == Popularization ==
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| The look-and-say sequence is also popularly known as the '''Morris Number Sequence''', after cryptographer [[Robert Morris (cryptographer)|Robert Morris]], and the puzzle 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>
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| == Computer program == | |
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| The following [[Python (programming language)|Python]] code generates a look-and-say sequence using [[lazy evaluation]]:
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| <source lang='python'> | |
| def look_and_say(member):
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| while True:
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| yield member
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| breakpoints = ([0] + [i for i in range(1, len(member))
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| if member[i - 1] != member[i]]
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| + [len(member)])
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| groups = [member[breakpoints[i - 1]:breakpoints[i]]
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| for i in range(1, len(breakpoints))]
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| member = ''.join(str(len(group)) + group[0] for group in groups)
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| # Print the 10-element sequence beginning with "1"
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| sequence = look_and_say("1")
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| for i in range(10):
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| print sequence.next()
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| </source> | |
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| The following [[JavaScript]] code uses a [[regular expression]] to generate the first 10 elements of a look-and-say sequence:
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| <source lang='javascript'>
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| var number = "1";
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| var regexp = /(\d)\1*/g;
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| for (var i = 0; i < 10; i++) {
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| console.log(number);
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| number = number.replace(regexp, function(fullmatch, symbol) {
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| return fullmatch.length.toString() + symbol;
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| });
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| }
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| </source>
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| ==Variations==
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| {{Unreferenced section|date=May 2012}}
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| 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, 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.
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| 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").
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| 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]].
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| ==See also==
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| * [[Kolakoski sequence]]
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| ==References==
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| <references/>
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| == External links ==
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| * {{MathWorld|urlname=LookandSaySequence|title=Look and Say Sequence}}
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| * [http://www.se16.info/js/looknsay.htm Look and Say sequence generator]
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| * {{SloanesRef |sequencenumber=A014715|name=Decimal expansion of Conway's constant}}
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| * [http://www.nitrxgen.net/source/lookandsay/ Look and Say sequence] - from 0 up to 70 (very large)
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| * [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]
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| [[Category:Base-dependent integer sequences]]
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| [[Category:Algebraic numbers]]
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| [[Category:Mathematical constants]]
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