A225224 -id:A225224 - cp's OEIS frontend

A005150 Look and Say sequence: describe the previous term! (method A - initial term is 1).

1, 11, 21, 1211, 111221, 312211, 13112221, 1113213211, 31131211131221, 13211311123113112211, 11131221133112132113212221, 3113112221232112111312211312113211, 1321132132111213122112311311222113111221131221, 11131221131211131231121113112221121321132132211331222113112211, 311311222113111231131112132112311321322112111312211312111322212311322113212221

Offset: 1

N. J. A. Sloane

Method A = "frequency" followed by "digit"-indication.
Also known as the "Say What You See" sequence.
Only the digits 1, 2 and 3 appear in any term. - Robert G. Wilson v, Jan 22 2004
All terms end with 1 (the seed) and, except the third a(3), begin with 1 or 3. - Jean-Christophe Hervé, May 07 2013
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
This sequence is called "suite de Conway" in French (see Wikipédia link). - Bernard Schott, Jan 10 2021
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 Belgrade, Yugoslavia. See the Editor's note on page 4, directly preceding Conway's article in Eureka referenced below. - Harlan J. Brothers, May 03 2024

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

John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 208.
S. R. Finch, Mathematical Constants, Cambridge, 2003, section 6.12 Conway's Constant, pp. 452-455.
M. Gilpin, On the generalized Gleichniszahlen-Reihe sequence, Manuscript, Jul 05 1994.
A. Lakhtakia and C. Pickover, Observations on the Gleichniszahlen-Reihe: An Unusual Number Theory Sequence, J. Recreational Math., 25 (No. 3, 1993), 192-198.
Clifford A. Pickover, Computers and the Imagination, St Martin's Press, NY, 1991.
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.
C. A. Pickover, The Math Book, Sterling, NY, 2009; see p. 486.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
James J. Tattersall, Elementary Number Theory in Nine Chapters, 1999, p. 23.
I. Vardi, Computational Recreations in Mathematica. Addison-Wesley, Redwood City, CA, 1991, p. 4.

T. D. Noe, Table of n, a(n) for n = 1..25
Henry Bottomley, Evolution of Conway's 92 Look and Say audioactive elements.
Éric Brier, Rémi Géraud-Stewart, David Naccache, Alessandro Pacco, and Emanuele Troiani, Stuttering Conway Sequences Are Still Conway Sequences, arXiv:2006.06837 [math.DS], 2020.
Éric Brier, Rémi Géraud-Stewart, David Naccache, Alessandro Pacco, and Emanuele Troiani, The Look-and-Say The Biggest Sequence Eventually Cycles, arXiv:2006.07246 [math.DS], 2020.
Onno M. Cain and Sela T. Enin, Inventory Loops (i.e. Counting Sequences) have Pre-period 2 max S_1 + 60, arXiv:2004.00209 [math.NT], 2020.
Ben Chen, Richard Chen, Joshua Guo, Tanya Khovanova, Shane Lee, Neil Malur, Nastia Polina, Poonam Sahoo, Anuj Sakarda, Nathan Sheffield, and Armaan Tipirneni, On Base 3/2 and its Sequences, arXiv:1808.04304 [math.NT], 2018.
J. H. Conway, The weird and wonderful chemistry of audioactive decay, Eureka 46 (1986) 5-16.
J. H. Conway, The weird and wonderful chemistry of audioactive decay, in T. M. Cover and Gopinath, eds., Open Problems in Communication and Computation, Springer, NY 1987, pp. 173-188.
J. H. Conway and Brady Haran, Look-and-Say Numbers (2014), Numberphile video.
S. B. Ekhad and D. Zeilberger, Proof of Conway's Lost Cosmological Theorem, arXiv:math/9808077 [math.CO], 1998.
S. B. Ekhad and D. Zeilberger, Proof of Conway's lost cosmological theorem, Electron. Res. Announc. Amer. Math. Soc. 3 (1997), 78-82.
S. Eliahou and M. J. Erickson, Mutually describing multisets and integer partitions, Discrete Mathematics, Volume 313, Issue 4, Feb 28 2013, Pages 422-433. - From _N. J. A. Sloane_, Jan 03 2013
S. R. Finch, Conway's Constant [From the Wayback Machine]
Steven Finch, The On-Line Encyclopedia of Integer Sequences, founded in 1964 by N. J. A. Sloane, A Tribute to John Horton Conway, The Mathematical Intelligencer (2021) Vol. 43, 146-147.
X. Gourdon and B. Salvy, Effective asymptotics of linear recurrences with rational coefficients, Discrete Mathematics, vol. 153, no. 1-3, 1996, pages 145-163. See p. 161.
M. Hilgemeier, Die Gleichniszahlen-Reihe, in Bild der Wissenschaft, 12 (1986), 194-195, with permission from the Konradin Medien GmbH.
M. Hilgemeier, One metaphor fits all, in Fractal Horizons, ed. C. A Pickover, St. Martins, NY, 1996, pp. 137-161.
R. A. Litherland, Conway's Cosmological Theorem (Overview).
R. A. Litherland, Conway's Cosmological Theorem, 12 pages, Apr 14 2006 (pdf file).
R. A. Litherland, Programs for Conway's Cosmological Theorem, (gzipped tar ball).
R. A. Litherland, The audioactive package.
M. Lothaire, Algebraic Combinatorics on Words, Cambridge, 2002, see p. 37, etc.
MacTutor History of Mathematics, John H. Conway
O. Martin, Look-and-Say Biochemistry: Exponential RNA and Multistranded DNA, Amer. Math. Monthly, 113 (No. 4, 2006), 289-307. - From _N. J. A. Sloane_, Feb 19 2013
Thomas Morrill, Look, Knave, arXiv:2004.06414 [math.CO], 2020.
Paulo Ortolan, Java program for A005150.
Matt Parker, Can you trust an elegant conjecture?, Stand-Up Maths, 2022, video.
Rosetta Code, Look and say sequence programs in over 60 languages.
J. Sauerberg and L. Shu, The long and the short on counting sequences, Amer. Math. Monthly, 104 (1997), 306-317.
T. Sillke, Conway sequence.
L. J. Upton, Letter to N. J. A. Sloane, Jan 08 1991.
Kevin Watkins, Abstract Interpretation Using Laziness: Proving Conway's Lost Cosmological Theorem.
Kevin Watkins, Proving Conway's Lost Cosmological Theorem, POP seminar talk, CMU, Dec 2006.
Eric Weisstein's World of Mathematics, Look and Say Sequence.
Wikipedia, Look-and-say sequence.
Wikipédia, Suite de Conway.
W. W. Zadrozny, Abstraction, Reasoning and Deep Learning: A Study of the "Look and Say" Sequence, arXiv:2109.12755 [cs.AI], 2022.
Julia Witte Zimmerman, Denis Hudon, Kathryn Cramer, Jonathan St. Onge, Mikaela Fudolig, Milo Z. Trujillo, Christopher M. Danforth, and Peter Sheridan Dodds, A blind spot for large language models: Supradiegetic linguistic information, arXiv:2306.06794 [cs.CL], 2023.
Julia Witte Zimmerman, Denis Hudon, Kathryn Cramer, Alejandro J. Ruiz, Calla Beauregard, Ashley Fehr, Mikaela Irene Fudolig, Bradford Demarest, Yoshi Meke Bird, Milo Z. Trujillo, Christopher M. Danforth, and Peter Sheridan Dodds, Tokens, the oft-overlooked appetizer: Large language models, the distributional hypothesis, and meaning, arXiv:2412.10924 [cs.CL], 2024. See pp. 21, 28.

Cf. A001155, A006751, A006715, A001140, A001141, A001143, A001145, A001151, A001154, A007651, A060857.
Cf. A001387, Periodic table: A119566.
Cf. A225224, A221646, A225212 (continuous versions).
Apart from the first term, all terms are in A001637.
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).
About primes: A079562 (number of distinct prime factors), A100108 (terms that are primes), A334132 (smallest prime factor).
Cf. A014715 (Conway's constant), A098097 (terms interpreted as written in base 4).

import List
say :: Integer -> Integer
say = read . concatMap saygroup . group . show
where saygroup s = (show $ length s) ++ [head s]
look_and_say :: [Integer]
look_and_say = 1 : map say look_and_say
-- Josh Triplett (josh(AT)freedesktop.org), Jan 03 2007

a005150 = foldl1 (\v d -> 10 * v + d) . map toInteger . a034002_row
-- Reinhard Zumkeller, Aug 09 2012

Java
```
See Paulo Ortolan link.
```

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} ]
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 *)

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

$str="1"; for (1 .. shift(@ARGV)) { print($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);
# Jeff Quilici (jeff(AT)quilici.com), Aug 12 2003

# This outputs the first n elements of the sequence, where n is given on the command line.
$s = 1;
for (2..shift @ARGV) {
print "$s, ";
$s =~ s/(.)\1*/(length $&).$1/eg;
}
# Arne 'Timwi' Heizmann (timwi(AT)gmx.net), Mar 12 2008
print "$s\n";

def A005150(n):
    p = "1"
    seq = [1]
    while (n > 1):
        q = ''
        idx = 0 # Index
        l = len(p) # Length
        while idx < l:
            start = idx
            idx = idx + 1
            while idx < l and p[idx] == p[start]:
                idx = idx + 1
            q = q + str(idx-start) + p[start]
        n, p = n - 1, q
        seq.append(int(p))
    return seq
# Olivier Mengue (dolmen(AT)users.sourceforge.net), Jul 01 2005

def A005150(n):
    seq = [1] + [None] * (n - 1) # allocate entire array space
    def say(s):
        acc = '' # initialize accumulator
        while len(s) > 0:
            i = 0
            c = s[0] # char of first run
            while (i < len(s) and s[i] == c): # scan first digit run
                i += 1
            acc += str(i) + c # append description of first run
            if i == len(s):
                break # done
            else:
                s = s[i:] # trim leading run of digits
        return acc
    for i in range(1, n):
        seq[i] = int(say(str(seq[i-1])))
    return seq
# E. Johnson (ejohnso9(AT)earthlink.net), Mar 31 2008

# program without string operations
def sign(n): return int(n > 0)
def say(a):
    r = 0
    p = 0
    while a > 0:
        c = 3 - sign((a % 100) % 11) - sign((a % 1000) % 111)
        r += (10 * c + (a % 10)) * 10**(2*p)
        a //= 10**c
        p += 1
    return r
a = 1
for i in range(1, 26):
    print(i, a)
    a = say(a)
# Volker Diels-Grabsch, Aug 18 2013

import re
def lookandsay(limit, sequence = 1):
    if limit > 1:
        return lookandsay(limit-1, "".join([str(len(match.group()))+match.group()[0] for matchNum, match in enumerate(re.finditer(r"(\w)\1*", str(sequence)))]))
    else:
        return sequence
# lookandsay(3) --> 21
# Nicola Vanoni, Nov 29 2016

import itertools
x = "1"
for i in range(20):
    print(x)
    x = ''.join(str(len(list(g)))+k for k,g in itertools.groupby(x))
# Matthew Cotton, Nov 12 2019

a(n+1) = A045918(a(n)). - Reinhard Zumkeller, Aug 09 2012
a(n) = Sum_{k=1..A005341(n)} A034002(n,k)*10^(A005341(n)-k). - Reinhard Zumkeller, Dec 15 2012
a(n) = A004086(A007651(n)). - Bernard Schott, Jan 08 2021
A055642(a(n+1)) = A005341(n+1) = 2*A043562(a(n)). - Ya-Ping Lu, Jan 28 2025
Conjecture: DC(a(n)) ~ k * (Conway's constant)^n where k is approximately 1.021... and DC denotes the number of digit changes in the decimal representation of n (e.g., DC(13112221)=4 because 1->3, 3-1, 1->2, 2->1). - Bill McEachen, May 09 2025
Conjecture: lim_{n->infinity} (c2+c3-c1)/(c1+c2+c3) = 0.01 approximately, where ci is the number of appearances of 'i' in a(n). - Ya-Ping Lu, Jun 05 2025

A221646 A continuous "Look and Say" sequence (without repetition, method 2).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 2, 2, 1, 2, 2, 1, 1, 2, 2, 2, 1, 3, 2, 1, 1, 1, 3, 1, 2, 3, 1, 1, 3, 1, 1, 1, 2, 1, 3, 2, 1, 1, 3, 3, 1, 1, 2, 1, 1, 1, 3, 1, 2, 2, 1, 2, 3, 2, 1, 1, 2, 3, 1, 1, 3, 1, 1, 2, 2, 1, 1, 1, 2, 1, 3, 1, 2, 2, 1, 1, 2, 1, 3, 2, 1, 1, 3, 2, 1, 2, 2

Offset: 1

Jean-Christophe Hervé, May 05 2013

A variant of Conway's 'Look-and-Say' sequence A005150, without run cut-off. It describes at each step the preceding digits taken altogether.
There are different optional rules to build such a sequence. This method 2 does not consider already said digits.
As in the original Look-and-Say sequence, a(n) is always equal to 1, 2 or 3. The subsequence 3,3,3 never appears.
The sequence is determined by pairs of digits. Terms of even rank are counts while terms of odd rank are figures.

			a(1) = 1, then a(2) = 1 and a(3) = 1 (one 1). Leaving out the first 1 already said, we now have two 1's, then a(4) = 2 and a(5) = 1, and then a(6) = 1, a(7) = 2, a(8) = 2, a(9) = 1, etc.

J.-C. Hervé, Table of n, a(n) for n = 1..10000

Cf. A005150 (original look-and-say sequence).

Cf. A225212, A225224 (other continuous variants).

/* computes first n terms in array a[] */
int *swys(int n) {
int a[n] ;
int see, say, c ;
a[0] = 1;
see = say = 0 ;
while( say < n-1 ) {
  c = 0 ;     /* count */
  dg = a[see] /* digit */
  while (see <= say) {
      if (a[see]== dg)  c += 1 ;
      else break ;
      see += 1 ;
      }
  a[++say] = c ;
  if (say < n-1) a[++say] = dg ;
  }
return(a);
}

n = 100; a[0] = 1; see = say = 0; While[say < n - 1, c = 0; dg = a[see]; While[see <= say, If[a[see] == dg, c += 1, Break[]]; see += 1]; a[++say] = c; If[say < n - 1, a[++say] = dg]]; Array[a, n, 0] (* Jean-François Alcover, Jul 11 2013, translated and adapted from J.-C. Hervé's C program *)

A225212 A "Look and Say" sequence (with nested repetitions).

Original entry on oeis.org

1, 1, 1, 3, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 1, 3, 2, 1, 1, 3, 2, 1, 3, 1, 1, 3, 1, 1, 1, 3, 2, 1, 1, 3, 2, 1, 1, 3, 2, 1, 1, 3, 3, 1, 1, 3, 1, 2, 2, 1, 1, 3, 1, 2, 1, 1, 3, 1, 1, 3, 1, 1, 1, 3, 2, 1, 1, 3, 2, 1, 1, 3, 2, 1, 1, 3, 3, 1, 1, 3, 1, 2, 2, 1

Offset: 1

Jean-Christophe Hervé, May 01 2013

A variant of the 'Look-and-Say' sequence A005150 that describes at each step the preceding digits altogether since the beginning. The sequence is built by blocks, each new block describing the preceding ones, always returning to the beginning: 1; 1,1; 3,1; 3,1,1,3,1,1; 3,1,1,3,1,1,1,3,2,1,1,3,2,1; etc. This generates indefinitely nested repeats: 31, 311311, 311311, ... The sequence A225224 is a variant that avoids these repetitions.
The size of the block of rank k equals 2^k for k = 1 or 2, and is < 2^k for any k >= 3.
Except the first two blocks, each block begins with 31 and ends with 1.
As in the original A005150, a(n) is always equal to 1, 2 or 3. The subsequence 3,3,3 never appears.

			a(1) = 1, you then see "one 1": a(2) = 1 (one) and a(3) = 1; Looking at a(1), a(2) and a(3) altogether, you then see "three 1" : a(4) = 3 and a(5) = 1. The sequence is then built by blocks, each new block describing the preceding ones since the beginning of the sequence: 1; 1,1; 3,1; 3,1,1,3,1,1; etc.

John Cerkan, Table of n, a(n) for n = 1..10000

Cf. A005150, A225224.

f[seq_] := Join[seq, {Length[#], First[#]}& /@ Split[seq]] // Flatten; Nest[f, {1}, 5] (* Jean-François Alcover, May 02 2013 *)

A225330 A continuous "look-and-repeat" sequence (method 1).

Original entry on oeis.org

1, 1, 1, 1, 4, 4, 1, 2, 2, 4, 1, 1, 1, 2, 2, 2, 1, 1, 4, 3, 3, 1, 3, 3, 2, 2, 2, 1, 1, 1, 4, 2, 2, 3, 1, 1, 1, 2, 2, 3, 3, 3, 2, 3, 3, 1, 1, 1, 4, 2, 2, 2, 1, 1, 3, 3, 3, 1, 2, 2, 2, 3, 3, 3, 1, 1, 2, 2, 2, 3, 3, 3, 1, 1, 1, 4

Offset: 1

Jean-Christophe Hervé, May 12 2013

A variant of the 'look-and-repeat' sequence A225329, without run cut-off. It describes at each step the preceding digits by repeating the frequency number.
The sequence is determined by triples of digits. The first two terms of a triple are the repeated frequency and the last term is the digit.
a(n) is always equal to 1, 2, 3, 4 or 5.
However, the occurrence of 4 is specific to this variant (method and seed), and only due to the initial sequence of four 1's. No other series of four identical digit happens in the sequence.
There are different optional rules to build such a sequence. This method 1 does not consider already said digits, unless if the length of the sequence of repeated figures to which they belongs change : this happens only once at the beginning, with the first 1 which is considered twice (and this brings up the 4): 1 ->  1,1,1 -> 4,4,1. The variant A225330 never considers already said digit (and does not contain 4). With other seeds (for example, 2 or 3), this special case at the beginning does not arise, and both variants coincide (and do not contain 4).
Applying the look-and-repeat principle to the sequence itself, it is simply shift four ranks to the left.

			a(1) = 1, you then see "one 1" and repeating "one", a(2) = a(3) = 1 (one) and a(4) = 1; Looking at a(1), a(2), a(3), and a(4) altogether, you then see "four 1": a(5) = a(6) = 4 and a(7) = 1, etc.

Cf. A225331 (a close variant), A225329 (look-and-repeat by block), A005150 (original look-and-say), A225224, A221646, A225212 (continuous look-and-say versions).

A088203 Infinite audioactive word that shifts 1 place left under "Look and Say" method A, starting with a(1)=2.

Original entry on oeis.org

2, 1, 2, 1, 1, 1, 2, 3, 1, 1, 2, 1, 3, 2, 1, 1, 2, 1, 1, 1, 3, 1, 2, 2, 1, 1, 2, 3, 1, 1, 3, 1, 1, 2, 2, 2, 1, 1, 2, 1, 3, 2, 1, 1, 3, 2, 1, 3, 2, 2, 1, 1, 2, 1, 1, 1, 3, 1, 2, 2, 1, 1, 3, 1, 2, 1, 1, 1, 3, 2, 2, 2, 1, 1, 2, 3, 1, 1, 3, 1, 1, 2, 2, 2, 1, 1, 3, 1, 1, 1, 2, 3, 1, 1, 3, 3, 2, 2, 1, 1, 2, 1, 3, 2, 1

Offset: 1

Paul D. Hanna, Sep 22 2003

A006751(n) = concatenation of n-th row. - Reinhard Zumkeller, Aug 09 2012
From Jean-Christophe Hervé, May 07 2013: (Start)
The sequence is obtained continuously by applying the look-and-say rule from seed 2: 2 -> 1,2 -> 1,1,1,2 -> etc. The sequence is then determined by pairs of digits. Terms of even ranks are counts while odd ranks are figures. A225224 and A221646 are from seed 1 and A088204 from seed 3.
The present sequence is the concatenation of A006751 (original look-and-say method by blocks) because, with seed 2, all blocks of A006751 begin with 1 or 3 and end with 2 and therefore, there is no possible interaction between blocks after concatenation. (End)

J. H. Conway, The weird and wonderful chemistry of audioactive decay, in T. M. Cover and Gopinath, eds., Open Problems in Communication and Computation, Springer, NY 1987, pp. 173-188.

Reinhard Zumkeller, Rows n = 1..25 of triangle, flattened
Kevin Watkins, Abstract Interpretation Using Laziness: Proving Conway's Lost Cosmological Theorem,
Kevin Watkins, Proving Conway's Lost Cosmological Theorem, POP seminar talk, CMU, Dec 2006
Eric Weisstein's World of Mathematics, Look and Say Sequence
Wikipedia, Look-and-say sequence

Cf. A088204, A087282, A005150.
Cf. A034002, A005150, A006751.
Cf. A225224, A221646 (seed one).

-- see Watkins link, p. 3.
import Data.List (group)
a088203 n k = a088203_tabf !! (n-1) !! (k-1)
a088203_row n = a088203_tabf !! (n-1)
a088203_tabf = iterate
               (concat . map (\xs -> [length xs, head xs]) . group) [2]
-- Reinhard Zumkeller, Aug 09 2012

A225329 Look-and-repeat: similar to look-and-say except frequency is repeated.

Original entry on oeis.org

1, 111, 331, 223111, 222113331, 332221333111, 223332111333331, 222333112331553111, 332333221112223111225113331, 223112333222331332113331222115221333111

Offset: 1

Jean-Christophe Hervé, May 12 2013

Repeated frequency followed by digit-indication. Repeating the frequency allows 5 to appear, in addition to 1, 2 and 3 which are already contained in Conway's original look-and-say sequence. However, 4 still does not appear.
The sequence is determined by triples of digits. The first two terms of a triple are the repeated figure and the last term is the digit.
Therefore, sequences of form xy (x != y), xxyy can never appear. A fortiori, the sequence never contains series of four identical digits, but contains series of five 3, which make appear the 5's (55 and 5). However five 5's never appear. Proof: suppose it appears for the first time in a(n)-a(n+4); because of 'five five 5' in 55555, it would imply that 55555 appears form a smaller n, which is a contradiction. By the same argument, 555 also never appear.
Also 22222 or 11111 are impossible : 22222 would imply a preceding 22yy and 11111 a preceding 1x (x != 1), but both cannot exist.
All terms end with 1 (the seed) and, except the first two, begin with 2 or 3.

			The term after 331 is obtained by saying (repeating) two two 3, one one 1, which gives 223111.

Cf. A005150 (original look-and-say), A225224, A221646, A225212 (continuous look-and-say versions), A225330, A225331 (continuous look-and-repeat).

A225331 A continuous "look-and-repeat" sequence (method 2).

Original entry on oeis.org

1, 1, 1, 1, 3, 3, 1, 2, 2, 3, 1, 1, 1, 2, 2, 2, 1, 1, 3, 3, 3, 1, 3, 3, 2, 2, 2, 1, 3, 3, 3, 1, 1, 1, 2, 2, 3, 3, 3, 2, 1, 1, 1, 3, 3, 3, 3, 3, 1, 2, 2, 2, 3, 3, 3, 1, 1, 2, 3, 3, 1, 5, 5, 3, 1, 1, 1, 3, 3, 2, 3, 3

Offset: 1

Jean-Christophe Hervé, May 12 2013

A variant of the 'look-and-repeat' sequence A225329, without run cut-off. It describes at each step the preceding digits by repeating the frequency number.
The sequence is determined by triples of digits. The first two terms of a triple are the repeated frequency and the last term is the digit.
There are different optional rules to build such a sequence. This method 2 never considers twice the already said digits.
With this rule and seed, a(n) is always equal to 1, 2, 3 or 5, and the sequence is the simple concatenation of the look-and-repeat sequence by block A225329. This is because all blocks of A225329 begin with 2 or 3 and end with 2 and therefore, there is no possible interaction between blocks after concatenation.
It never contains runs of exactly four identical digits (except the first four ones), but it does contain runs of five identical digits. However, five 5's never appear. Proof: suppose '55555' appears for the first time in a(n)..a(n+4); because of 'five five 5' in 55555, it would imply that 55555 appears from a smaller n, which is a contradiction.

			a(1) = 1, then a(2) = a(3) = a(4) = 1 (one one 1). Leaving out the first 1 already said, we now have three 1's, then a(5) = a(6) = 3, and a(7) = 1, etc.

Cf. A225330 (a close variant with 4's), A225329 (look-and-repeat by block), A005150 (original look-and-say), A225224, A221646, A225212 (continuous look-and-say versions).

A088204 Infinite audioactive word that shifts 1 place left under "Look and Say" method A, starting with a(1)=3.

Original entry on oeis.org

3, 1, 3, 1, 1, 1, 3, 3, 1, 2, 3, 1, 1, 1, 2, 1, 3, 3, 1, 1, 2, 1, 1, 2, 3, 2, 1, 1, 2, 2, 1, 1, 2, 1, 3, 1, 2, 2, 1, 2, 2, 2, 1, 1, 2, 1, 1, 1, 3, 1, 1, 2, 2, 1, 1, 3, 2, 2, 1, 1, 2, 3, 1, 1, 3, 2, 1, 2, 2, 2, 1, 1, 3, 2, 2, 2, 1, 1, 2, 1, 3, 2, 1, 1, 3, 1, 2, 1, 1, 3, 2, 2, 1, 1, 3, 3, 2, 2, 1, 1, 2, 1, 1, 1, 3

Offset: 1

Paul D. Hanna, Sep 22 2003

The sequence is obtained continuously by applying the look-and-say rule from seed 3 : 3 -> 1,3 -> 1,1,1,3 -> etc. The sequence is then determined by pairs of digits. Terms of even ranks are counts while odd ranks are figures. A225224 and A221646 are from seed 1 and A088203 from seed 2. [Jean-Christophe Hervé, May 07 2013]

Cf. A088203, A087282, A005150, A006715.

Cf. A225224, A221646 (seed one).

A225332 A continuous "look-and-repeat" sequence (seed 2).

Original entry on oeis.org

2, 1, 1, 2, 2, 2, 1, 3, 3, 2, 1, 1, 1, 2, 2, 3, 1, 1, 2, 3, 3, 1, 2, 2, 2, 1, 1, 3, 2, 2, 1, 1, 1, 2, 2, 2, 3, 1, 1, 1, 3, 3, 2, 2, 2, 1, 1, 1, 3, 2, 2, 2, 3, 3, 1, 3, 3, 2, 1, 1, 3, 3, 3, 1, 2, 2, 3, 3, 3, 2, 3, 3

Offset: 1

Jean-Christophe Hervé, May 12 2013

The 'look-and-repeat' sequence A225330, with seed 2. The variant A225331 with the same seed 2 gives this same sequence.
It describes at each step the preceding digits by repeating the frequency number.
The sequence is determined by triples of digits. The first two terms of a triple are the repeated count and the last term is the digit.
a(n) is always equal to 1, 2, 3, or 5. No series of four identical digits happens in the sequence, nor any of five 5's.
Applying the look-and-repeat principle to the sequence itself, it is simply shift one rank to the left.

			a(1) = 2, you then see "one 2" and repeating "one", a(2) = a(3) = 1 (one) and a(4) = 2; you have then two 1's, so 2, 2, 1; then three 2, so 3, 3, 1, etc.

Cf. A225330, A225331 (two variants with seed 1), A225329 (look-and-repeat by block), A005150 (original look-and-say), A225224, A221646, A225212 (continuous look-and-say versions).

A225333 Look-and-repeat: similar to look-and-say except frequency is repeated (seed 2).

Original entry on oeis.org

2, 112, 221112, 222331112, 332223331112, 223332333331112, 222333112553331112, 332333221112225333331112, 223112333222331332115553331112, 222113221112333332223111223112221335333331112

Offset: 1

Jean-Christophe Hervé, May 12 2013

Look-and-repeat sequence A225329 with seed 2.
Contains 1, 2, 3 and 5, but not 4.
All terms end with 2 (the seed) and, starting at the fourth, with 3331112, which makes the 5 appear.
All terms except the second begin with 2 or 3; this is a direct consequence of the look-and-repeat rule.

			The term after 112 is obtained by saying (repeating) two two 1, one one 2, which gives 221112.

Cf. A225329 (seed one), A225332 (continuous look-and-repeat, seed 2), A225330, A225331 (continuous look-and-repeat, seed 1).

Cf. A005150 (original look-and-repeat), A225224, A221646, A225212 (continuous look-and-repeat versions).

cp's OEIS Frontend

A005150 Look and Say sequence: describe the previous term! (method A - initial term is 1).

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A221646 A continuous "Look and Say" sequence (without repetition, method 2).

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A225212 A "Look and Say" sequence (with nested repetitions).

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A225330 A continuous "look-and-repeat" sequence (method 1).

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A088203 Infinite audioactive word that shifts 1 place left under "Look and Say" method A, starting with a(1)=2.

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A225329 Look-and-repeat: similar to look-and-say except frequency is repeated.

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A225331 A continuous "look-and-repeat" sequence (method 2).

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A088204 Infinite audioactive word that shifts 1 place left under "Look and Say" method A, starting with a(1)=3.

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