A007651 -id:A007651 - cp's OEIS frontend

A005341 Length of n-th term in Look and Say sequences A005150 and A007651.

1, 2, 2, 4, 6, 6, 8, 10, 14, 20, 26, 34, 46, 62, 78, 102, 134, 176, 226, 302, 408, 528, 678, 904, 1182, 1540, 2012, 2606, 3410, 4462, 5808, 7586, 9898, 12884, 16774, 21890, 28528, 37158, 48410, 63138, 82350, 107312, 139984, 182376, 237746, 310036, 403966, 526646, 686646

Offset: 1

Jeffrey Shallit

Row lengths of A034002 and of A220424. - Reinhard Zumkeller, Dec 15 2012

Satisfies a recurrence of order 72. The characteristic polynomial of this recurrence is a degree-72 polynomial that factors as (x-1)*q(x), where q(x) is a degree-71 polynomial. The unique positive real root of q is approximately 1.3036 and is called Conway's constant (A014715), which equals the limiting ratio a(n+1)/a(n). - Nathaniel Johnston, Apr 12 2018 [Corrected by Richard Stanley, Dec 26 2018]

J. H. Conway, The weird and wonderful chemistry of audioactive decay, in T. M. Cover and Gopinath, eds., Open Problems in Communication and Communications, Springer, NY 1987, pp. 173-188.
S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 452-455.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Peter J. C. Moses, Table of n, a(n) for n = 1..3000 (first 71 terms from Zak Seidov)
S. R. Finch, Conway's Constant.
S. R. Finch, Conway's Constant. [From the Wayback Machine]
Christoph Koutschan,Regular Languages and their Generating Functions: The Inverse Problem.
Eric Weisstein's World of Mathematics, Look and Say Sequence.

a005341 = length . a034002_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[ Length[ F[ n ] ], {n, 1, 51} ]
p = {12, -18, 18, -18, 18, -20, -22, 31, 15, -4, -4, -19, 62, -50, -21, -11, 41, 54, -56, -44, 15, -27, -15, 45, -8, 89, -64, -66, -25, 38, 126, -39, -32, -33, -65, 107, 14, 16, -13, -79, 7, 42, 12, 8, -26, -9, 35, -23, -20, -30, 34, 58, -1, -20, -36, -6, 13, 8, 6, 3, -1, -4, -1, -4, -5, -1, 8, 6, 0, -6, -4, 1, 0, 1, 1, 1, 1, -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 23 2013 *)

print1(a=1);for(i=2,100,print1(",",#Str(a=A005150(2,a))))  \\ M. F. Hasler, Nov 08 2011

a(n) = A055642(A005150(n)) = A055642(A007651(n)). - Reinhard Zumkeller, Dec 15 2012

More terms from Mike Keith

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]

A022466 Number of 1's in n-th term of A007651.

Original entry on oeis.org

1, 2, 1, 3, 4, 3, 4, 6, 8, 12, 13, 18, 24, 33, 39, 52, 67, 88, 113, 155, 211, 264, 331, 455, 596, 762, 1000, 1288, 1688, 2222, 2884, 3754, 4915, 6364, 8300, 10845, 14138, 18372, 23966, 31254, 40821, 53113, 69336, 90273, 117678, 153518, 199990, 260671, 340007

Offset: 1

Clark Kimberling

nonn

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

Cf. A007651.

p={10,-3,15,-13,26,-10,-18,-33,21,-3,9,-22,40,8,12,-52,-19,73,38,5,14,-25,-35,-20,-25,53,29,4,-91,-43,42,18,36,-5,-99,10,7,55,44,-44,-53,-5,15,47,-19,-23,13,21,18,-55,-39,15,35,36,-7,-23,-9,-3,3,2,1,2,6,6,-2,-9,-5,2,7,3,-1,-1,-2,-2,-1,0,2,1}; q={-6,3,-6,12,-4,7,-7,1,0,5,-2,-4,-12,2,7,12,-7,-10,-4,3,9,-7,0,-8,14,-3,9,2,-3,-10,-2,-6,1,10,-3,1,7,-7,7,-12,-5,8,6,10,-8,-8,-7,-3,9,1,6,6,-2,-3,-10,-2,3,5,2,-1,-1,-1,-1,-1,1,2,2,-1,-2,-1,0,1}; 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 *)

A022467 Number of 2's in n-th term of A007651.

Original entry on oeis.org

0, 0, 1, 1, 2, 2, 3, 2, 3, 4, 8, 10, 14, 18, 24, 28, 41, 56, 69, 91, 123, 164, 221, 286, 366, 494, 636, 832, 1095, 1423, 1847, 2440, 3154, 4166, 5366, 7025, 9119, 11925, 15495, 20259, 26320, 34409, 44827, 58432, 76193, 99353, 129347, 168789, 219947, 286629

Offset: 1

Clark Kimberling

nonn

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

p={-12,20,-11,8,9,-30,39,-58,67,-63,29,-14,0,39,15,-63,-11,30,31,22,-11,-21,23,-46,34,-46,60,26,-64,20,-77,59,41,-7,-32,-14,-6,9,5,16,1,33,-35,0,-64,38,25,38,3,-49,-27,1,11,21,2,-1,13,-9,-14,-11,1,14,11,2,-6,-8,-5,2,5,3,1,-1,-1,-2,0,0,1,0,0}; 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 *)

A022468 Number of 3's in n-th term of A007651.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 2, 3, 4, 5, 6, 8, 11, 15, 22, 26, 32, 44, 56, 74, 100, 126, 163, 220, 284, 376, 486, 627, 817, 1077, 1392, 1829, 2354, 3108, 4020, 5271, 6861, 8949, 11625, 15209, 19790, 25821, 33671, 43875, 57165, 74629, 97186, 126692, 165238

Offset: 1

Clark Kimberling

nonn

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

p={10,-15,11,-18,12,14,-25,12,-28,43,-13,2,0,-21,10,10,3,8,-10,-11,5,9,-18,16,-31,35,-20,15,-6,-10,36,-44,9,-1,3,16,-11,23,-3,-25,-17,-27,43,24,24,-33,-24,-7,14,6,9,-5,10,0,-9,-9,-12,7,14,7,-1,-9,-6,2,3,2,1,-1,0,-1,-1,0,0,1,0,0,0,0,0}; 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

A022470 Describe the previous term! (method B - initial term is 2).

Original entry on oeis.org

2, 21, 2111, 2113, 211231, 2112213111, 211222113113, 21122312311231, 2112223111213112213111, 21122331132111311222113113, 211222321231211331122312311231, 21122331211121311121123212223111213112213111, 21122232112113211131132112213121112331132111311222113113

Offset: 1

Clark Kimberling

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

			E.g., the term after 2113 is obtained by saying "2 once, 1 twice, 3 once", which gives 211231.

Peter J. C. Moses, Table of n, a(n) for n = 1..23

Cf. A007651, A022499, A022500-A022505.

Cf. A006751 (method A).

a[1] = 2; a[n_] := a[n] = FromDigits[Flatten[{First[#], Length[#]} & /@ Split[IntegerDigits[a[n - 1]]]]]; Map[a,Range[1, 23]] (* 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 aupton(nn): return list(accumulate(repeat(2, nn), summarize))
print(aupton(13)) # Michael S. Branicky, Feb 21 2021

A022499 Describe the previous term! (method B - initial term is 3).

Original entry on oeis.org

3, 31, 3111, 3113, 311231, 3112213111, 311222113113, 31122312311231, 3112223111213112213111, 31122331132111311222113113, 311222321231211331122312311231

Offset: 1

N. J. A. Sloane

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

			E.g. the term after 3113 is obtained by saying "3 once, 1 twice, 3 once", which gives 311231.

Seiichi Manyama, Table of n, a(n) for n = 1..23

Cf. A007651, A022470, A022500-A022505.

Cf. A006715.

a[1] = 3;
a[n_] := a[n] = {#[[1]], Length[#]}& /@ Split[a[n-1] // IntegerDigits] // Flatten // FromDigits;
Array[a, 11] (* Jean-François Alcover, Jul 13 2016, updated Jan 05 2018 *)

A022500 Describe the previous term! (method B - initial term is 4).

Original entry on oeis.org

4, 41, 4111, 4113, 411231, 4112213111, 411222113113, 41122312311231, 4112223111213112213111, 41122331132111311222113113, 411222321231211331122312311231

Offset: 1

N. J. A. Sloane

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

			E.g. the term after 4113 is obtained by saying "4 once, 1 twice, 3 once", which gives 411231.

Seiichi Manyama, Table of n, a(n) for n = 1..23

Cf. A007651, A022470, A022499, A022501-A022505.

a[1] = 4;
a[n_] := a[n] = {#[[1]], Length[#]}& /@ Split[a[n-1] // IntegerDigits] // Flatten // FromDigits;
Array[a, 11] (* Jean-François Alcover, Jul 13 2016 *)

A022505 Describe the previous term! (method B - initial term is 9).

Original entry on oeis.org

9, 91, 9111, 9113, 911231, 9112213111, 911222113113, 91122312311231, 9112223111213112213111, 91122331132111311222113113, 911222321231211331122312311231

Offset: 1

N. J. A. Sloane

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

			E.g., the term after 9113 is obtained by saying "9 once, 1 twice, 3 once", which gives 911231.

Cf. A007651, A022470, A022499, A022500, A022501, A022502, A022503, A022504.

a(12)-(14) needlessly added by Alvin Hoover Belt, Jan 31 2010

cp's OEIS Frontend

A005341 Length of n-th term in Look and Say sequences A005150 and A007651.

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

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A022466 Number of 1's in n-th term of A007651.

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A022467 Number of 2's in n-th term of A007651.

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A022468 Number of 3's in n-th term of A007651.

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

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A022470 Describe the previous term! (method B - initial term is 2).

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