%I M4780 #268 May 04 2024 17:51:45

%S 1,11,21,1211,111221,312211,13112221,1113213211,31131211131221,

%T 13211311123113112211,11131221133112132113212221,

%U 3113112221232112111312211312113211,1321132132111213122112311311222113111221131221,11131221131211131231121113112221121321132132211331222113112211,311311222113111231131112132112311321322112111312211312111322212311322113212221

%N Look and Say sequence: describe the previous term! (method A - initial term is 1).

%C Method A = "frequency" followed by "digit"-indication.

%C Also known as the "Say What You See" sequence.

%C Only the digits 1, 2 and 3 appear in any term. - _Robert G. Wilson v_, Jan 22 2004

%C All terms end with 1 (the seed) and, except the third a(3), begin with 1 or 3. - _Jean-Christophe Hervé_, May 07 2013

%C Proof that 333 never appears in any a(n): suppose it appears for the first time in a(n); because of "three 3" in 333, it would imply that 333 is also in a(n-1), which is a contradiction. - _Jean-Christophe Hervé_, May 09 2013

%C This sequence is called "suite de Conway" in French (see Wikipédia link). - _Bernard Schott_, Jan 10 2021

%C Contrary to many accounts (including an earlier comment on this page), Conway did not invent the sequence. The first mention of the sequence appears to date back to the 1977 International Mathematical Olympiad in Belgrave, Yugoslavia. See the Editor's note on page 4, directly preceding Conway's article in Eureka referenced below. - _Harlan J. Brothers_, May 03 2024

%D S. R. Finch, Mathematical Constants, Cambridge, 2003, section 6.12 Conway's Constant, pp. 452-455.

%D M. Gilpin, On the generalized Gleichniszahlen-Reihe sequence, Manuscript, Jul 05 1994.

%D A. Lakhtakia and C. Pickover, Observations on the Gleichniszahlen-Reihe: An Unusual Number Theory Sequence, J. Recreational Math., 25 (No. 3, 1993), 192-198.

%D Clifford A. Pickover, Computers and the Imagination, St Martin's Press, NY, 1991.

%D Clifford A. Pickover, Fractal horizons: the future use of fractals, New York: St. Martin's Press, 1996. ISBN 0312125992. Chapter 7 has an extensive description of the elements and their properties.

%D C. A. Pickover, The Math Book, Sterling, NY, 2009; see p. 486.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%D James J. Tattersall, Elementary Number Theory in Nine Chapters, 1999, p. 23.

%D I. Vardi, Computational Recreations in Mathematica. Addison-Wesley, Redwood City, CA, 1991, p. 4.

%H T. D. Noe, <a href="/A005150/b005150.txt">Table of n, a(n) for n = 1..25</a>

%H Henry Bottomley, <a href="http://www.se16.info/js/lands2.htm">Evolution of Conway's 92 Look and Say audioactive elements</a>.

%H Éric Brier, Rémi Géraud-Stewart, David Naccache, Alessandro Pacco, and Emanuele Troiani, <a href="https://arxiv.org/abs/2006.06837">Stuttering Conway Sequences Are Still Conway Sequences</a>, arXiv:2006.06837 [math.DS], 2020.

%H Éric Brier, Rémi Géraud-Stewart, David Naccache, Alessandro Pacco, and Emanuele Troiani, <a href="https://arxiv.org/abs/2006.07246">The Look-and-Say The Biggest Sequence Eventually Cycles</a>, arXiv:2006.07246 [math.DS], 2020.

%H Onno M. Cain and Sela T. Enin, <a href="https://arxiv.org/abs/2004.00209">Inventory Loops (i.e. Counting Sequences) have Pre-period 2 max S_1 + 60</a>, arXiv:2004.00209 [math.NT], 2020.

%H Ben Chen, Richard Chen, Joshua Guo, Tanya Khovanova, Shane Lee, Neil Malur, Nastia Polina, Poonam Sahoo, Anuj Sakarda, Nathan Sheffield, and Armaan Tipirneni, <a href="https://arxiv.org/abs/1808.04199">On Base 3/2 and its Sequences</a>, arXiv:1808.04304 [math.NT], 2018.

%H J. H. Conway, <a href="https://static01.nyt.com/packages/pdf/crossword/GENIUS_AT_PLAY_Eureka_Article.pdf">The weird and wonderful chemistry of audioactive decay</a>, Eureka 46 (1986) 5-16.

%H J. H. Conway, <a href="http://dx.doi.org/10.1007/978-1-4612-4808-8_53">The weird and wonderful chemistry of audioactive decay</a>, in T. M. Cover and Gopinath, eds., Open Problems in Communication and Computation, Springer, NY 1987, pp. 173-188.

%H J. H. Conway and Brady Haran, <a href="https://www.youtube.com/watch?v=ea7lJkEhytA">Look-and-Say Numbers</a> (2014), Numberphile video.

%H S. B. Ekhad and D. Zeilberger, <a href="https://arxiv.org/abs/math/9808077">Proof of Conway's Lost Cosmological Theorem</a>, arXiv:math/9808077 [math.CO], 1998.

%H S. B. Ekhad and D. Zeilberger, <a href="http://www.ams.org/era/1997-03-11/S1079-6762-97-00026-7/home.html">Proof of Conway's lost cosmological theorem</a>, Electron. Res. Announc. Amer. Math. Soc. 3 (1997), 78-82.

%H S. Eliahou and M. J. Erickson, <a href="https://doi.org/10.1016/j.disc.2012.11.014">Mutually describing multisets and integer partitions</a>, Discrete Mathematics, Volume 313, Issue 4, Feb 28 2013, Pages 422-433. - From _N. J. A. Sloane_, Jan 03 2013

%H S. R. Finch, <a href="http://web.archive.org/web/20010207194413 /http://www.mathsoft.com/asolve/constant/cnwy/cnwy.html">Conway's Constant</a> [From the Wayback Machine]

%H Steven Finch, <a href="https://doi.org/10.1007/s00283-021-10060-2">The On-Line Encyclopedia of Integer Sequences, founded in 1964 by N. J. A. Sloane</a>, A Tribute to John Horton Conway, The Mathematical Intelligencer (2021) Vol. 43, 146-147.

%H X. Gourdon and B. Salvy, <a href="https://doi.org/10.1016/0012-365X(95)00133-H">Effective asymptotics of linear recurrences with rational coefficients</a>, Discrete Mathematics, vol. 153, no. 1-3, 1996, pages 145-163. See p. 161.

%H M. Hilgemeier, <a href="/A005150/a005150_1.pdf">Die Gleichniszahlen-Reihe</a>, in Bild der Wissenschaft, 12 (1986), 194-195, with permission from the Konradin Medien GmbH.

%H M. Hilgemeier, <a href="http://www.se16.info/mhi/">One metaphor fits all</a>, in Fractal Horizons, ed. C. A Pickover, St. Martins, NY, 1996, pp. 137-161.

%H R. A. Litherland, <a href="/A005150/a005150.html">Conway's Cosmological Theorem (Overview)</a>.

%H R. A. Litherland, <a href="/A005150/a005150_3.pdf">Conway's Cosmological Theorem</a>, 12 pages, Apr 14 2006 (pdf file).

%H R. A. Litherland, <a href="/A005150/a005150.tar.gz">Programs for Conway's Cosmological Theorem</a>, (gzipped tar ball).

%H R. A. Litherland, <a href="/A005150/a005150_4.pdf">The audioactive package</a>.

%H M. Lothaire, <a href="http://www-igm.univ-mlv.fr/~berstel/Lothaire/">Algebraic Combinatorics on Words</a>, Cambridge, 2002, see p. 37, etc.

%H MacTutor History of Mathematics, <a href="https://mathshistory.st-andrews.ac.uk/Biographies/Conway/">John H. Conway</a>

%H O. Martin, <a href="http://www.jstor.org/stable/27641915">Look-and-Say Biochemistry: Exponential RNA and Multistranded DNA</a>, Amer. Math. Monthly, 113 (No. 4, 2006), 289-307. - From _N. J. A. Sloane_, Feb 19 2013

%H Thomas Morrill, <a href="https://arxiv.org/abs/2004.06414">Look, Knave</a>, arXiv:2004.06414 [math.CO], 2020.

%H Paulo Ortolan, <a href="/A005150/a005150.txt">Java program for A005150</a>.

%H Matt Parker, <a href="https://www.youtube.com/watch?v=EGoRJePORHs">Can you trust an elegant conjecture?</a>, Stand-Up Maths, 2022, video.

%H Rosetta Code, <a href="http://rosettacode.org/wiki/Look-and-say_sequence">Look and say sequence</a> programs in over 60 languages.

%H J. Sauerberg and L. Shu, <a href="http://www.jstor.org/stable/2974579">The long and the short on counting sequences</a>, Amer. Math. Monthly, 104 (1997), 306-317.

%H T. Sillke, <a href="http://www.mathematik.uni-bielefeld.de/~sillke/SEQUENCES/series000">Conway sequence</a>.

%H L. J. Upton, <a href="/A005151/a005151.pdf">Letter to N. J. A. Sloane</a>, Jan 08 1991.

%H Kevin Watkins, <a href="http://www.cs.cmu.edu/~kw/pubs/conway.pdf">Abstract Interpretation Using Laziness: Proving Conway's Lost Cosmological Theorem</a>.

%H Kevin Watkins, <a href="http://www.cs.cmu.edu/~kw/pubs/conwayslides.pdf">Proving Conway's Lost Cosmological Theorem</a>, POP seminar talk, CMU, Dec 2006.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LookandSaySequence.html">Look and Say Sequence</a>.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Look-and-say_sequence">Look-and-say sequence</a>.

%H Wikipédia, <a href="https://fr.wikipedia.org/wiki/Suite_de_Conway">Suite de Conway</a>.

%H W. W. Zadrozny, <a href="https://arxiv.org/abs/2109.12755">Abstraction, Reasoning and Deep Learning: A Study of the "Look and Say" Sequence</a>, arXiv:2109.12755 [cs.AI], 2022.

%H Julia Witte Zimmerman, Denis Hudon, Kathryn Cramer, Jonathan St. Onge, Mikaela Fudolig, Milo Z. Trujillo, Christopher M. Danforth, and Peter Sheridan Dodds, <a href="https://arxiv.org/abs/2306.06794">A blind spot for large language models: Supradiegetic linguistic information</a>, arXiv:2306.06794 [cs.CL], 2023.

%F a(n+1) = A045918(a(n)). - _Reinhard Zumkeller_, Aug 09 2012

%F a(n) = Sum_{k=1..A005341(n)} A034002(n,k)*10^(A005341(n)-k). - _Reinhard Zumkeller_, Dec 15 2012

%F a(n) = A004086(A007651(n)). - _Bernard Schott_, Jan 08 2021

%e The term after 1211 is obtained by saying "one 1, one 2, two 1's", which gives 111221.

%t RunLengthEncode[ x_List ] := (Through[ {First, Length}[ #1 ] ] &) /@ Split[ x ]; LookAndSay[ n_, d_:1 ] := NestList[ Flatten[ Reverse /@ RunLengthEncode[ # ] ] &, {d}, n - 1 ]; F[ n_ ] := LookAndSay[ n, 1 ][ [ n ] ]; Table[ FromDigits[ F[ n ] ], {n, 1, 15} ]

%t A005150[1] := 1; A005150[n_] := A005150[n] = FromDigits[Flatten[{Length[#], First[#]}&/@Split[IntegerDigits[A005150[n-1]]]]]; Map[A005150, Range[25]] (* _Peter J. C. Moses_, Mar 21 2013 *)

%o (Haskell)

%o import List

%o say :: Integer -> Integer

%o say = read . concatMap saygroup . group . show

%o where saygroup s = (show $ length s) ++ [head s]

%o look_and_say :: [Integer]

%o look_and_say = 1 : map say look_and_say

%o -- Josh Triplett (josh(AT)freedesktop.org), Jan 03 2007

%o (Haskell)

%o a005150 = foldl1 (\v d -> 10 * v + d) . map toInteger . a034002_row

%o -- _Reinhard Zumkeller_, Aug 09 2012

%o (Java) See Paulo Ortolan link.

%o (Perl)

%o $str="1"; for (1 .. shift(@ARGV)) { print($str, ","); @a = split(//,$str); $str=""; $nd=shift(@a); while (defined($nd)) { $d=$nd; $cnt=0; while (defined($nd) && ($nd eq $d)) { $cnt++; $nd = shift(@a); } $str .= $cnt.$d; } } print($str);

%o # Jeff Quilici (jeff(AT)quilici.com), Aug 12 2003

%o (Perl)

%o # This outputs the first n elements of the sequence, where n is given on the command line.

%o $s = 1;

%o for (2..shift @ARGV) {

%o print "$s, ";

%o $s =~ s/(.)\1*/(length $&).$1/eg;

%o }

%o # Arne 'Timwi' Heizmann (timwi(AT)gmx.net), Mar 12 2008

%o print "$s\n";

%o (Python)

%o def A005150(n):

%o p = "1"

%o seq = [1]

%o while (n > 1):

%o q = ''

%o idx = 0 # Index

%o l = len(p) # Length

%o while idx < l:

%o start = idx

%o idx = idx + 1

%o while idx < l and p[idx] == p[start]:

%o idx = idx + 1

%o q = q + str(idx-start) + p[start]

%o n, p = n - 1, q

%o seq.append(int(p))

%o return seq

%o # Olivier Mengue (dolmen(AT)users.sourceforge.net), Jul 01 2005

%o (Python)

%o def A005150(n):

%o seq = [1] + [None] * (n - 1) # allocate entire array space

%o def say(s):

%o acc = '' # initialize accumulator

%o while len(s) > 0:

%o i = 0

%o c = s[0] # char of first run

%o while (i < len(s) and s[i] == c): # scan first digit run

%o i += 1

%o acc += str(i) + c # append description of first run

%o if i == len(s):

%o break # done

%o else:

%o s = s[i:] # trim leading run of digits

%o return acc

%o for i in range(1, n):

%o seq[i] = int(say(str(seq[i-1])))

%o return seq

%o # E. Johnson (ejohnso9(AT)earthlink.net), Mar 31 2008

%o (Python)

%o # program without string operations

%o def sign(n): return cmp(n, 0)

%o def say(a):

%o r = 0

%o p = 0

%o while a > 0:

%o c = 3 - sign((a % 100) % 11) - sign((a % 1000) % 111)

%o r += (10 * c + (a % 10)) * 10**(2*p)

%o a /= 10**c

%o p += 1

%o return r

%o a = 1

%o for i in range(1, 26):

%o print(i, a)

%o a = say(a)

%o # _Volker Diels-Grabsch_, Aug 18 2013

%o (Python)

%o import re

%o def lookandsay(limit, sequence = 1):

%o if limit > 1:

%o return lookandsay(limit-1, "".join([str(len(match.group()))+match.group()[0] for matchNum, match in enumerate(re.finditer(r"(\w)\1*", str(sequence)))]))

%o else:

%o return sequence

%o # lookandsay(3) --> 21

%o # _Nicola Vanoni_, Nov 29 2016

%o (Python)

%o import itertools

%o x = "1"

%o for i in range(20):

%o print(x)

%o x = ''.join(str(len(list(g)))+k for k,g in itertools.groupby(x))

%o # _Matthew Cotton_, Nov 12 2019

%o (PARI) A005150(n,a=1)={ while(n--, my(c=1); for(j=2,#a=Vec(Str(a)), if( a[j-1]==a[j], a[j-1]=""; c++, a[j-1]=Str(c,a[j-1]); c=1)); a[#a]=Str(c,a[#a]); a=concat(a)); a } \\ _M. F. Hasler_, Jun 30 2011

%Y Cf. A001155, A006751, A006715, A001140, A001141, A001143, A001145, A001151, A001154, A007651, A060857.

%Y Cf. A001387, Periodic table: A119566.

%Y Cf. A225224, A221646, A225212 (continuous versions).

%Y Apart from the first term, all terms are in A001637.

%Y About digits: A005341 (number of digits), A022466 (number of 1's), A022467 (number of 2's), A022468 (number of 3's), A004977 (sum of digits), A253677 (product of digits).

%Y About primes: A079562 (number of distinct prime factors), A100108 (terms that are primes), A334132 (smallest prime factor).

%Y Cf. A014715 (Conway's constant), A098097 (terms interpreted as written in base 4).

%K nonn,base,easy,nice

%O 1,2

%A _N. J. A. Sloane_