A005341 -id:A005341 - cp's OEIS frontend

A056635 Difference between length (A005341) and sum of digits (A004977) of n-th term in Look and Say Sequence (A005150).

0, 0, 1, 1, 2, 4, 5, 6, 9, 12, 18, 22, 30, 40, 54, 72, 93, 120, 157, 203, 271, 364, 473, 612, 806, 1062, 1388, 1804, 2349, 3057, 4001, 5224, 6812, 8874, 11582, 15065, 19661, 25647, 33393, 43509, 56738, 73989, 96469, 125774, 163943, 213683, 278605

Offset: 1

Robert G. Wilson v, Aug 08 2000

Peter J. C. Moses, Table of n, a(n) for n = 1..1000

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[ Apply[ Plus, F[ n ] ]-Length[ F[ n ] ], {n, 1, 53} ] (* Eric Weisstein *)
p={8,-10,11,-28,33,-2,-11,-34,11,23,3,-10,0,-3,35,-43,-5,46,11,0,-1,-3,-13,-14,-28,24,20,56,-76,0,-5,-29,59,-9,-26,18,-28,55,-1,-34,-33,-21,51,48,-16,-28,-23,24,31,-37,-9,-9,31,21,-16,-19,-11,5,14,3,-1,-4,-1,6,0,-4,-3,0,5,1,-1,-1,-1,0,0,0,1,0,0}; q={6,-9,9,-18,16,-11,14,-8,1,-5,7,2,8,-14,-5,-5,19,3,-6,-7,-6,16,-7,8,-22,17,-12,7,5,7,-8,4,-7,-9,13,-4,-6,14,-14,19,-7,-13,2,-4,18,0,-1,-4,-12,8,-5,0,8,1,7,-8,-5,-2,3,3,0,0,0,0,-2,-1,0,3,1,-1,-1,-1,1}; gf=Fold[x #1+#2&,0,p]/Fold[x #1+#2&,0,q]; CoefficientList[Series[gf,{x,0,99}],x] (* Peter J. C. Moses, Jun 24 2013 *)

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

A007651 Describe the previous term! (method B - initial term is 1).

Original entry on oeis.org

1, 11, 12, 1121, 122111, 112213, 12221131, 1123123111, 12213111213113, 11221131132111311231, 12221231123121133112213111, 1123112131122131112112321222113113, 1221311221113112221131132112213121112312311231

Offset: 1

N. J. A. Sloane

Method B = 'digit'-indication followed by 'frequency'.

			The term after 1121 is obtained by saying "1 twice, 2 once, 1 once", which gives 122111.

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

Reinhard Zumkeller, Table of n, a(n) for n = 1..25

Cf. A005150, A022470, A022499, A022500-A022505.

a007651 = foldl1 (\v d -> 10 * v + d) . map toInteger . a220424_row
-- Reinhard Zumkeller, Dec 15 2012

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[ Reverse[ F[ n ] ] ], {n, 1, 15} ]
a[1] = 1; a[n_] := a[n] = FromDigits[Flatten[{First[#], Length[#]}&/@Split[IntegerDigits[a[n-1]]]]]; Map[a, Range[25]] (* Peter J. C. Moses, Mar 22 2013 *)

from itertools import accumulate, groupby, repeat
def summarize(n, _): return int("".join(k+str(len(list(g))) for k, g in groupby(str(n))))
def aupto(terms): return list(accumulate(repeat(1, terms), summarize))
print(aupto(13)) # Michael S. Branicky, Sep 18 2022

a(n) = Sum_{k=1..A005341(n)} A220424(n,k)*10^(A005341(n)-k). - Reinhard Zumkeller, Dec 15 2012

A034002 A005150 expanded into single digits.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

			Initial rows                             A005150
  1:  1                                           1
  2:  1,1                                        11
  3:  2,1                                        21
  4:  1,2,1,1                                  1211
  5:  1,1,1,2,2,1                            111221
  6:  3,1,2,2,1,1                            312211
  7:  1,3,1,1,2,2,2,1                      13112221
  8:  1,1,1,3,2,1,3,2,1,1                1113213211
  9:  3,1,1,3,1,2,1,1,1,3,1,2,2,1    31131211131221

Reinhard Zumkeller, Rows n = 1..25 of triangle, flattened
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. DOI: 10.1007/978-1-4612-4808-8_53.
M. Lothaire, Algebraic Combinatorics on Words, Cambridge, 2002, see p. 36.
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

See the entry for A005150 for much more about this sequence.
Cf. A088203.
Cf. A005341 (row lengths), A220424 (method B version).

-- see Watkins link, p. 3.
import Data.List (group)
a034002 n k = a034002_tabf !! (n-1) !! (k-1)
a034002_row n = a034002_tabf !! (n-1)
a034002_tabf = iterate
               (concat . map (\xs -> [length xs, head xs]) . group) [1]
-- Reinhard Zumkeller, Aug 09 2012

from sympy import flatten
l=[1]
L=[1]
n=s=1
y=''
while n<21:
    x=str(l[n - 1]) + ' '
    for i in range(len(x) - 1):
        if x[i]==x[i + 1]: s+=1
        else:
            y+=str(s)+str(x[i])
            s=1
    x=''
    n+=1
    l.append(int(y))
    L.append([int(a) for a in list(y)])
    y=''
    s=1
print(l) # A005150
print(flatten(L)) # Indranil Ghosh, Jul 05 2017

A005150(n) = Sum_{k=1..A005341(n)} T(n,k)*10^(A005341(n) - k). - Reinhard Zumkeller, Dec 15 2012

Offset changed and keyword tabf added by Reinhard Zumkeller, Aug 09 2012

A004977 Sum of digits of n-th term in Look and Say sequence A005150.

Original entry on oeis.org

1, 2, 3, 5, 8, 10, 13, 16, 23, 32, 44, 56, 76, 102, 132, 174, 227, 296, 383, 505, 679, 892, 1151, 1516, 1988, 2602, 3400, 4410, 5759, 7519, 9809, 12810, 16710, 21758, 28356, 36955, 48189, 62805, 81803, 106647, 139088, 181301, 236453, 308150, 401689

Offset: 1

Clark Kimberling

It appears that the ratio of consecutive terms approaches Conway's constant 1.303.. (A014715). The terms divided by the numbers of added digits also would tend to a constant, i.e. A004977(n)/A005341(n)->const. If the digits in A005150 occur with constant probabilities c1, c2, c3 then A004977(n)=A005341(n)*(c1+2*c2+3*c3) and this conjecture entails the convergences noted here. - Alexandre Losev, Aug 31 2005

Peter J. C. Moses, Table of n, a(n) for n = 1..1000
Albert Frank, International Contest Of Logical Sequences, 2002 - 2003. Item 9
Albert Frank, Solutions of International Contest Of Logical Sequences, 2002 - 2003.

Cf. A005150.

Cf. A005150, A005341, A014715.

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[ Apply[ Plus, F[ n ] ], {n, 1, 51} ]
p={-4,8,-7,-10,15,18,11,-65,-4,27,7,9,-62,47,56,-32,-46,-8,67,44,-16,24,2,-59,-20,-65,84,122,-51,-38,-131,10,91,24,39,-89,-42,39,12,45,-40,-63,39,40,10,-19,-58,47,51,-7,-43,-67,32,41,20,-13,-24,-3,8,0,0,0,0,10,5,-3,-11,-6,5,7,3,-2,-1,-1,-1,-1,0,1,1}; q={6,-9,9,-18,16,-11,14,-8,1,-5,7,2,8,-14,-5,-5,19,3,-6,-7,-6,16,-7,8,-22,17,-12,7,5,7,-8,4,-7,-9,13,-4,-6,14,-14,19,-7,-13,2,-4,18,0,-1,-4,-12,8,-5,0,8,1,7,-8,-5,-2,3,3,0,0,0,0,-2,-1,0,3,1,-1,-1,-1,1}; gf=Fold[x #1+#2&,0,p]/Fold[x #1+#2&,0,q]; CoefficientList[Series[gf,{x,0,99}],x] (* Peter J. C. Moses, Jun 24 2013 *)

A220424 Triangle read by rows: A007651 expanded into single digits.

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, Dec 15 2012

A007651(n) = sum{T(n,k)*10^(A005341(n)-k): k=1..A005341(n)}.

			.  Initial rows                          A007651
.  1:  1                                           1
.  2:  1,1                                        11
.  3:  1,2                                        12
.  4:  1,1,2,1                                  1121
.  5:  1,2,2,1,1,1                            122111
.  6:  1,1,2,2,1,3                            112213
.  7:  1,2,2,2,1,1,3,1                      12221131
.  8:  1,1,2,3,1,2,3,1,1,1                1123123111
.  9:  1,2,2,1,3,1,1,1,2,1,3,1,1,3    12213111213113 .

Reinhard Zumkeller, Rows n = 1..25 of triangle, flattened
Eric Weisstein's World of Mathematics, Look and Say Sequence
Wikipedia, Look-and-say sequence

Cf. A005341 (row lengths), A034002 (method A version).

import Data.List (group)
a220424 n k = a220424_tabf !! (n-1) !! (k-1)
a220424_row n = a220424_tabf !! (n-1)
a220424_tabf = iterate
               (concatMap (\xs -> [head xs, length xs]) . group) [1]

A121993 Numbers k that yield a smaller number a(k) under the "Look and Say" function A045918.

Original entry on oeis.org

33, 44, 55, 66, 77, 88, 99, 111, 222, 333, 444, 555, 666, 777, 888, 999, 1111, 1222, 1333, 1444, 1555, 1666, 1777, 1888, 1999, 2000, 2111, 2222, 2233, 2244, 2255, 2266, 2277, 2288, 2299, 2333, 2444, 2555, 2666, 2777, 2888, 2999, 3000, 3111, 3222, 3300, 3311

Offset: 1

Sergio Pimentel, Sep 11 2006

			a(26)=2000 because under the Look and Say operator, 2000 is described as one two three zeros or: 1230, which is smaller than 2000.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Look and Say Sequence.

Cf. A045918, A005150, A005341, A023989, A079475.

a121993 n = a121993_list !! (n-1)
a121993_list = filter (\x -> a045918 x < x) [0..]
-- Reinhard Zumkeller, Jan 25 2014

from itertools import groupby
def ok(n): return n > int(''.join(str(len(list(g)))+k for k, g in groupby(str(n))))
print([k for k in range(3312) if ok(k)]) # Michael S. Branicky, May 26 2023

A179999 Length of the n-th term in the modified Look and Say sequence A110393.

Original entry on oeis.org

1, 2, 2, 4, 6, 8, 10, 14, 18, 24, 30, 40, 50, 66, 82, 108, 134, 176, 218, 286, 354, 464, 574, 752, 930, 1218, 1506, 1972, 2438, 3192, 3946, 5166, 6386, 8360, 10334, 13528, 16722, 21890, 27058, 35420, 43782, 57312, 70842, 92734, 114626, 150048

Offset: 1

Nathaniel Johnston, Jan 13 2011

The average multiplicative growth from the n-th term to the (n+1)-st term is sqrt(phi) = 1.272..., where phi = (1+sqrt(5))/2 is the golden ratio, see A139339.

			The 6th term in A110393 is 21112211, so a(6) = 8.

Colin Barker, Table of n, a(n) for n = 1..1000
N. Johnston, Further Variants of the “Look-and-Say” Sequence
Index entries for linear recurrences with constant coefficients, signature (1,1,-1,1,-1).

Cf. A005341, A049194, A098596, A110393.

Cf. A001622, A134972, A139339, A239798.

CoefficientList[Series[((1+x) (-1-x+x^2) (1-x+x^2))/((1-x) (-1+x^2+x^4)),{x,0,99}],x] (* Peter J. C. Moses, Jun 23 2013 *)

Vec(x*(1 + x)*(1 + x - x^2)*(1 - x + x^2) / ((1 - x)*(1 - x^2 - x^4)) + O(x^50)) \\ Colin Barker, Aug 10 2019

a(n) = length(A110393(n)).
From Colin Barker, Aug 10 2019: (Start)
G.f.: x*(1 + x)*(1 + x - x^2)*(1 - x + x^2) / ((1 - x)*(1 - x^2 - x^4)).
a(n) = a(n-1) + a(n-2) - a(n-3) + a(n-4) - a(n-5) for n>6. (End)
From A.H.M. Smeets, Aug 10 2019 (Start)
Limit_{n->oo} a(n+1)/a(n) = (1+phi)/2 = (3+sqrt(5))/4 = A239798 for odd n.
Limit_{n->oo} a(n+1)/a(n) = 2/phi = 4/(1+sqrt(5)) = A134972 for even n.
Limit_{n->oo} a(n+2)/a(n) = (1+phi)/phi = phi = A001622. (End)
For odd n > 1, a(n) = 4*Fibonacci((n + 1)/2) - 2. For even n, a(n) = 2*Fibonacci(n/2 + 2) - 2. - Ehren Metcalfe, Aug 10 2019

A121994 Smallest natural number that yields a sequence of n decreasing numbers under the "Look and Say" operator A045918.

Original entry on oeis.org

1, 33, 333, 333111, 33333333333333333333333333333333311111111111

Offset: 0

Sergio Pimentel, Sep 11 2006

a(5) <= 33333333333333333333333333333333 '3's concatenated with 1111111111 '1's. - Tyler Busby, Feb 07 2023

			a(3)=333111 because under the Look and Say operator sequence A045918 it yields: 3331, 3311, 2321 which are all decreasing (3 in a row). The next term would be 12131211 which is greater than 2321.

Eric Weisstein's World of Mathematics, Look and Say.

Cf. A045918, A005150, A005341, A023989, A079475.

a(4) from Sergio Pimentel, Mar 05 2008

a(4) corrected by Tyler Busby, Feb 07 2023

cp's OEIS Frontend

A056635 Difference between length (A005341) and sum of digits (A004977) of n-th term in Look and Say Sequence (A005150).

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

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A007651 Describe the previous term! (method B - initial term is 1).

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A034002 A005150 expanded into single digits.

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A004977 Sum of digits of n-th term in Look and Say sequence A005150.

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A220424 Triangle read by rows: A007651 expanded into single digits.

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A121993 Numbers k that yield a smaller number a(k) under the "Look and Say" function A045918.

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