Look-and-say sequence

File:Conway constant.png. Conway's constant is marked with the Greek letter lambda (λ).]]

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 degenerate sequence: 22, 22, 22, 22, ... which remains the same size.

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.

{{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

}}

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 elements 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.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. 78–82. Retrieved July 4, 2011. Below is a table of all such elements:

The terms eventually grow in length by about 30% per generation. In particular, if L_n denotes the number of digits of the n-th member of the sequence, then the limit of the ratio $\backslash frac\{L\_\{n\; +\; 1\}\}\{L\_n\}$ exists and is given by

$$\backslash lim\_\{n\; \backslash to\; \backslash infty\}\; \backslash frac\{L\_\{n+1\}\}\{L\_\{n\}\}\; =\; \backslash lambda$$

where λ = 1.303577269034... {{OEIS|id=A014715}} is an algebraic number of degree 71. This fact was proven by Conway, and the constant λ 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 is the unique positive real root of the following polynomial {{OEIS|id=A137275}}:

$$\backslash begin\{matrix\}$$

& &\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}

This polynomial was correctly given in Conway's original Eureka article,

but in the reprinted version in the book edited by Cover and Gopinath the term $x^\{35\}$ was incorrectly printed with a minus sign in front.

{{Cite book

| last = Vardi

| first = Ilan

| title = Computational Recreations in Mathematica

| publisher = Addison-Wesley

| year = 1991

| isbn = 0-201-52989-0

}}

The look-and-say sequence is also popularly known as the Morris Number Sequence, after 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.[http://jamesthornton.com/fun/robert-morris-sequence.html Robert Morris Sequence], jamesthornton.com[https://web.archive.org/web/20110803133359/http://www.ocf.berkeley.edu/~stoll/number_sequence.html FAQ about Morris Number Sequence], ocf.berkeley.edu

{{Unreferenced section|date=May 2022}}

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.{{cite arXiv |last1=Kowacs |first1=André |title=Studies on the Pea Pattern Sequence |date=2017 |class=math.HO |eprint=1708.06452 }}{{cite journal |last1=Dassow |first1=J. |last2=Marcus |first2=S. |last3=Paun |first3=G. |title=Iterative reading of numbers and "black-holes" |journal=Periodica Mathematica Hungarica |date=1 October 1993 |volume=27 |issue=2 |pages=137–152 |doi=10.1007/BF01876638}}{{verify source|date=April 2025}} So 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 "atomic" terms 23322114 and 32232114. Since the sequence is infinite, the length of each element in the sequence is bounded, and there are only finitely many words that are at most a predetermined length, it must eventually repeat, and as a consequence, pea pattern sequences are always eventually periodic.

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 {{OEIS|id=A005151}}. 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"). This variation ultimately ends up repeating the number 21322314 ("two 1s, three 2s, two 3s and one 4").

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 {{nobr| 2 × Radix + 2 digits}} (22 digits for decimal: {{nobr|radix {{=}} 10}}) and may only exceed {{nobr| 3 × Radix digits}} (30 digits for decimal radix) in length for long, degenerate, initial seeds (sequence of "100 ones", etc.). For these extreme cases, individual elements of decimal sequences immediately settle into a permutation of the form {{nobr|{{math| a0 b1 c2 d3 e4 f5 g6 h7 i8 j9 }} }} where here the letters {{math| a–j }} are placeholders for digit counts from the preceding sequence element.

• Gijswijt's sequence
• Kolakoski sequence
• Autogram

{{reflist|group=note}}

• [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