A001387 -id:A001387 - cp's OEIS frontend

A049194 Number of digits in n-th term of A001387.

1, 2, 3, 6, 8, 12, 18, 27, 39, 58, 85, 125, 183, 269, 394, 578, 847, 1242, 1820, 2668, 3910, 5731, 8399, 12310, 18041, 26441, 38751, 56793, 83234, 121986, 178779, 262014, 384000, 562780, 824794, 1208795, 1771575, 2596370, 3805165, 5576741

Offset: 1

Olivier Gérard

Peter A. Hendriks, "A binary variant of Conway's audioactive sequence", lecture at 1192nd meeting of WWWW, Groningen, The Netherlands (Jul 15 1999).

Edward Krogius, Table of n, a(n) for n = 1..1000
T. Sillke, The binary form of Conway's sequence
Index entries for linear recurrences with constant coefficients, signature (1,1,0,0,-1).

Cf. A001387, A001609.

CoefficientList[Series[(1+x+x^3-x^4-x^5)/(1-x-x^2+x^5),{x,0,50}],x] (* Peter J. C. Moses, Jun 21 2013 *)

a(n) = if (n==3, 3, if (n==4, 6, floor((8/9 + (1/18)*(748 - 36*sqrt(93))^(1/3) + (1/18)*(748 + 36*sqrt(93))^(1/3)) * (1/3 + (1/6)*(116 - 12*sqrt(93))^(1/3) + (1/6)*(116 + 12*sqrt(93))^(1/3))^(n-1)))) \\ Michel Marcus, Mar 04 2013

a(n) = my(v=vector(n), u=[1,2,3,6]); if(n<=4, u[n], for(i=1, 4, v[i]=u[i]); for(i=5, n, v[i]=v[i-1]+v[i-3]+!(i%2)); v[n]) \\ Jianing Song, Apr 28 2019

a(n) = (8/9 + (1/18)*(748 - 36*sqrt(93))^(1/3) + (1/18)*(748 + 36*sqrt(93))^(1/3)) * (1/3 + (1/6)*(116 - 12*sqrt(93))^(1/3) + (1/6)*(116 + 12*sqrt(93))^(1/3))^(n-1).
The number of digits is equal to c*l^n rounded down to the nearest integer, where c and l are the real roots of 3x^3 - 8x^2 + 5x - 1 and x^3 - x^2 - 1 respectively, for all n except n = 2 and n = 3.
From Jianing Song, Apr 28 2019: (Start)
a(n) = a(n-1) + a(n-2) - a(n-5) for n >= 7. [Derived from the T. Sillke link above.]
a(n) = a(n-1) + a(n-3) if n is odd, a(n-1) + a(n-3) + 1 if n is even, n >= 5 (this does not hold for n = 4).
Limit_{n->oo} a(n)/A001609(n) = c, where c = 1.276742... is the unique real root of 3x^3 - 4x^2 + x - 1. (End)

More terms and formulas supplied by Gerton Lunter (gerton(AT)math.rug.nl)

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

Original entry on oeis.org

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

A049190 Start with 1. Convert to base 2, describe it in base 2, convert to base 10. Repeat with the new result.

Original entry on oeis.org

1, 3, 5, 59, 245, 2491, 235253, 127756731, 330567489269, 258479716298484155, 36823182192123209878050549, 25576412117054296344209353299113896379, 10994511204169842163496446583221775727830456269734123253

Offset: 1

Olivier Gérard

a(n) is A001387(n) converted to base 10. - Nathan Fox, Mar 07 2018

"Describe" means to apply the "look-and-say" function (cf. A045918), but the "count" is again expressed in binary (and concatenated with the digit), cf. examples. - M. F. Hasler, Jul 12 2025

			1 -> one 1 -> 11 -> 3;
3 -> 11 -> two 1s -> 101 -> 5;
5 -> 101 -> one 1, one 0, one 1 -> 111011 -> 59;
etc.

Cf. A005150, A001387.

A049190_first(N=13)=vector(N, i, N=if(i>1, my(d=binary(N), j=0); d=concat(d[^1]-d[^-1],-1); fromdigits(concat([concat(binary(-j+j=n), d[n]<0) | n<-[1..#d], d[n]]), 2), 1)) \\ M. F. Hasler, Jul 12 2025

Definition and Example corrected by Nathan Fox, Mar 07 2018

A001391 To get the 3rd term, for example, note that 2nd term has three (11 in binary!) 1's and one (1) 0, giving 11 1 1 0.

Original entry on oeis.org

10, 1110, 11110, 100110, 1110010110, 111100111010110, 100110011110111010110, 1110010110010011011110111010110, 1111001110101100111001011010011011110111010110

Offset: 1

N. J. A. Sloane, Thomas L. York

John Cerkan, Table of n, a(n) for n = 1..17

Cf. A001387.

A049064 Describe the previous term in binary (method A - initial term is 0).

Original entry on oeis.org

0, 10, 1110, 11110, 100110, 1110010110, 111100111010110, 100110011110111010110, 1110010110010011011110111010110, 1111001110101100111001011010011011110111010110, 1001100111101110101100111100111010110111001011010011011110111010110

Offset: 1

Olivier de Mouzon

Method A = 'frequency' (in binary mode) followed by 'digit'-indication.

The number of digits of a(n) is A001609(n) except for n = 2. See the link from T. Sillke below. - Jianing Song, Mar 16 2019

			E.g., the term after 11110 is obtained by saying "four (i.e., 100 in binary mode) 1, one 0", which gives 100110.

Kade Robertson, Table of n, a(n) for n = 1..18
Kade Robertson, Table of n, a(n) for n = 1..31
T. Sillke, The binary form of Conway's sequence

Cf. A001387 (initial term is 1), A001391, A001609 (number of digits), A259710 (written in decimal).

Decimal look-and-say sequences: A005150, A006751, A006715, A001140, A001141, A001143, A001145, A001151, A001154.

a(n) = A001391(n-1), n > 1. - R. J. Mathar, Oct 15 2008

Edited by Charles R Greathouse IV, Apr 06 2010
a(11) from Kade Robertson, Jun 24 2015
Offset corrected by Jianing Song, Mar 16 2019

A320890 a(1) = 11. For all subsequent terms a(n), take a(n-1) and substitute for the k-th digit the binary number of times that digit has appeared in a(n-1), reading left to right from the 1st to k-th digit.

Original entry on oeis.org

11, 110, 1101, 110111, 110111100101, 11011110010111010111111001000, 1101111001011101011111100100010011010101101111011001101111011111000010001111100010010100110101011

Offset: 1

Thomas Anton, Oct 23 2018

Each term is an initial segment of all of its successors.
There are always more 1's than 0's in a term.
The proportion of 0's or 1's in the n-th term approaches 1/2 as n approaches infinity.
Starting with any binary integer apart from 0 or 1 and applying the same process to yield a sequence s(n), we have that, for a sufficiently large x, a(n) is always an initial segment of s(n+x). The constancy and uniqueness of the limiting behavior of initial segments in base 2 is unique among all bases, unless the tally system is considered as a degenerate case.

			a(1) = 11
The first 1 is replaced with 1, and the second 1 is replaced with 10 (two), so a(2) = 110 (1|10)
The first 1 is replaced with 1, the second 1 with 10, and the first 0 with 1, so a(3) = 1101 (1|10|1)
The first 1 is replaced with 1, the second 1 with 10, the first 0 with 1, and the third 1 with 11 (three), so a(4) = 110111 (1|10|1|11)
The first 1 is replaced with 1, the second 1 with 10, the first 0 with 1, the third 1 with 11, the fourth 1 with 100, and the fifth 1 with 101, so a(5) = 110111100101 (1|10|1|11|100|101)
The first 1 is replaced with 1, the second 1 with 10, the first 0 with 1, the third 1 with 11, the fourth 1 with 100, the fifth 1 with 101, the sixth 1 with 110, the second 0 with 10, the third 0 with 11, the seventh 1 with 111, the fourth 0 with 100, and the eighth 1 with 1000, so a(6) = 11011110010111010111111001000 (1|10|1|11|100|101|110|10|11|111|100|1000)

Chai Wah Wu, Table of n, a(n) for n = 1..8

Cf. A005150, A007651, A001387, A005151.

FromDigits /@ Nest[Append[#, Flatten[IntegerDigits[#, 2] & /@ Table[Count[#, Last@ #] &@ #[[1 ;; k]], {k, Length@ #}]] &[#[[-1]] ] ] &, {{1, 1}}, 6] (* Michael De Vlieger, Oct 23 2018 *)

eva(n) = subst(Pol(n), x, 10)
replace(v) = my(w=[], zeros=0, ones=0); for(k=1, #v, if(v[k]==0, zeros++; w=concat(w, binary(zeros))); if(v[k]==1, ones++; w=concat(w, binary(ones)))); w
terms(n) = my(v=[1, 1], i=0); while(i < n, print1(eva(v), ", "); i++; v=replace(v))
/* Print initial 7 terms as follows: */
terms(7) \\ Felix Fröhlich, Oct 23 2018

A320890_list = [11]
while len(A320890_list)<10:
    a0,a1,s = 0,0,''
    for d in str(A320890_list[-1]):
        if d == '0':
            a0 += 1
            s += bin(a0)[2:]
        else:
            a1 += 1
            s += bin(a1)[2:]
    A320890_list.append(int(s)) # Chai Wah Wu, Nov 30 2018

A356329 Binary Look and Say sequence (method B - initial term is 1).

Original entry on oeis.org

1, 11, 110, 11001, 11001011, 1100101101110, 11001011011100111101, 11001011011100111101011000111, 1100101101110011110101100011101110011111, 1100101101110011110101100011101110011111011110101101

Offset: 1

Szumi Xie, Sep 18 2022

Method B = 'digit'-indication followed by 'frequency' in binary.
It is not true that every term is a prefix of the following term; the first counterexample is at the 15th term.
It appears that the lengths of the common prefixes between adjacent terms strictly increase.

			The term after 11001 is obtained by saying "1 twice (10 in binary), 0 twice (10) and 1 once (1)", giving 1 10 0 10 1 1.

Szumi Xie, Table of n, a(n) for n = 1..24

Cf. A001387, A007651.

import Data.List (group)
import Numeric (showBin)
a356329 n = a356329_list !! (n - 1)
a356329_list = read <$> iterate (concat . concatMap (\x -> [take 1 x, showBin (length x) ""]) . group) "1" :: [Integer]

a[1]=1; a[n_] := a[n]=Block[{s = Split@ IntegerDigits@ a[n-1]}, FromDigits@ Flatten@ Transpose[{First /@ s, IntegerDigits[ Length /@ s, 2]}]]; Array[a, 10] (* Giovanni Resta, Oct 05 2022 *)

from itertools import accumulate, groupby, repeat
def summarize(n, _): return int("".join(k+bin(len(list(g)))[2:] for k, g in groupby(str(n))))
def aupto(terms): return list(accumulate(repeat(1, terms), summarize))
print(aupto(11)) # Michael S. Branicky, Sep 18 2022

A363054 Look and say sequence: describe the previous term (method A, starting with 20).

Original entry on oeis.org

20, 1210, 11121110, 31123110, 132112132110, 11131221121113122110, 311311222112311311222110, 1321132132211213211321322110, 11131221131211132221121113122113121113222110, 3113112221131112311332211231131122211311123113322110

Offset: 1

Julia Zimmerman, May 15 2023

			The term after 1210 is given by saying "I see one 1, one 2, one 1, and one 0", and then writing down the digits as 11-12-11-10, yielding 11121110.

Michael De Vlieger, Table of n, a(n) for n = 1..21
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.

Other look-and-say sequences: A005150, A001155, A006751, A006715, A001140, A001141, A001143, A001145, A001151, A007651, A001387, A098595, A045918.

NestList[FromDigits@ Flatten@ Map[Reverse@ Tally[#][[1]] &, Split@ IntegerDigits[#] ] &, 20, 12] (* Michael De Vlieger, Jul 05 2023 *)

from itertools import count, groupby, islice
def LS(n): return int(''.join(str(len(list(g)))+k for k, g in groupby(str(n))))
def agen(an=20): yield an; yield from (an:=LS(an) for n in count(1))
print(list(islice(agen(), 10))) # Michael S. Branicky, May 15 2023

A260387 Numbers n = d_0d_1...d_n (n < 10) such that d_i is the number of digits equal to i in n (base b), where b is less than 10.

Original entry on oeis.org

12, 13, 320, 3201, 72200, 89000, 132110, 345000, 643000, 2320200, 3121300, 10103111, 11300130, 42430000, 51340000, 64030000, 72300000, 86300000, 125102000, 130213000, 211220001, 220101111, 323111000, 431130000, 614110000, 667000000, 2153100000, 2521002000, 3021211100

Offset: 1

Pieter Post, Jul 24 2015

The only terms having the same number of digits as the base are 13, 10103111, 211220001 and 220101111. For example, 13 is 1101_2,  which has 1 zero and 3 ones.
The least term with 10 digits that describes itself is 2153100000.
2153100000 is 104233022322_7, so it has 2 zeros, 1 one, 5 twos, 3 threes, 1 four, 0 fives, 0 sixes, 0 sevens, 0 eights and 0 nines in base 7.

			12 = 110_3, which has 1 zero and 2 ones.
13 = 1101_2, which has 1 zero and 3 ones.
320 = 11000_4, which has 3 zeros, 2 ones and 0 twos.
3201 = 100301_5, which has 3 zeros, 2 ones, 0 twos and 1 three.
72200 = 10200001002_3
89000 = 10101101110101000_2
132110 = 13211420_5
345000 = 122112020210_3
643000 1012200000211_3
42430000 = 2201312320300_4
51340000 = 3003312023200_4
64030000 = 3310100110300_4
72300000 = 122002100000_5
86300000 = 20000101111100022_3
431130000 = 110440340120_6
614110000 = 2224203010000_5
667000000 = 1201111002002222201_3
2153100000 = 104233022322_7

Cf. A001155, A001387, A005150, A046043, A138480.

a(10)-a(13), a(19)-a(23), a(28)-a(29) added by Giovanni Resta, Jul 26 2015

cp's OEIS Frontend

A049194 Number of digits in n-th term of A001387.

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

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A049190 Start with 1. Convert to base 2, describe it in base 2, convert to base 10. Repeat with the new result.

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A001391 To get the 3rd term, for example, note that 2nd term has three (11 in binary!) 1's and one (1) 0, giving 11 1 1 0.

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A049064 Describe the previous term in binary (method A - initial term is 0).

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A320890 a(1) = 11. For all subsequent terms a(n), take a(n-1) and substitute for the k-th digit the binary number of times that digit has appeared in a(n-1), reading left to right from the 1st to k-th digit.

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A356329 Binary Look and Say sequence (method B - initial term is 1).

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