A006751 -id:A006751 - cp's OEIS frontend

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

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

A001154 Describe the previous term! (method A - initial term is 9).

Original entry on oeis.org

9, 19, 1119, 3119, 132119, 1113122119, 311311222119, 13211321322119, 1113122113121113222119, 31131122211311123113322119, 132113213221133112132123222119

Offset: 1

N. J. A. Sloane

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

a(n+1) - a(n) is divisible by 10^5 for n > 5. - Altug Alkan, Dec 04 2015

			E.g. the term after 3119 is obtained by saying "one 3, two 1's, one 9", which gives 132119.

S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 452-455.
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..20
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.
S. R. Finch, Conway's Constant [Broken link]
S. R. Finch, Conway's Constant [From the Wayback Machine]

Cf. A001155, A005150, A006751, A006715, A001140, A001141, A001143, A001145, A001151.

RunLengthEncode[x_List] := (Through[{First, Length}[ #1]] &) /@ Split[x]; LookAndSay[n_, d_: 1] := NestList[Flatten[Reverse /@ RunLengthEncode[ # ]] &, {d}, n - 1]; F[n_] := LookAndSay[n, 9][[n]]; Table[FromDigits[F[n]], {n, 1, 11}] (* Zerinvary Lajos, Jul 08 2009 *)

A138484 Say what you see in previous term, from the right, reporting total number for each digit encountered. Initial term is 0.

Original entry on oeis.org

0, 10, 1011, 3110, 102113, 13311210, 10411223, 1322311410, 1041142322, 3213243110, 1031331422, 2214313310, 1031331422, 2214313310, 1031331422, 2214313310, 1031331422, 2214313310, 1031331422, 2214313310, 1031331422

Offset: 0

Paolo P. Lava and Giorgio Balzarotti, Mar 20 2008

After a while sequence has period 2 -> {1031331422,2214313310}

			To get the term after 102113, we say: one 3's, three 1's, one 2's, one 0's, so 13311210.

Cf. A063850, A022482, A005150, A005151, A006751, A006715, A006711, A022506-A022513, A138485-A138493.

A138493 Say what you see in previous term, from the right, reporting total number for each digit encountered. Initial term is 9.

Original entry on oeis.org

9, 19, 1911, 3119, 192113, 13311219, 19411223, 1322311419, 1941142322, 3213243119, 1931331422, 2214313319, 1931331422, 2214313319, 1931331422, 2214313319, 1931331422, 2214313319, 1931331422, 2214313319, 1931331422

Offset: 0

Paolo P. Lava and Giorgio Balzarotti, Mar 20 2008

After a while sequence has period 2 -> {1931331422,2214313319}

			To get the term after 192113, we say: one 3's, three 1's, one 2's, one 9's, so 13311219

Cf. A063850, A022482, A005150, A005151, A006751, A006715, A006711, A022506-A022513, A138484-A138492.

A014715 Decimal expansion of Conway's constant.

Original entry on oeis.org

1, 3, 0, 3, 5, 7, 7, 2, 6, 9, 0, 3, 4, 2, 9, 6, 3, 9, 1, 2, 5, 7, 0, 9, 9, 1, 1, 2, 1, 5, 2, 5, 5, 1, 8, 9, 0, 7, 3, 0, 7, 0, 2, 5, 0, 4, 6, 5, 9, 4, 0, 4, 8, 7, 5, 7, 5, 4, 8, 6, 1, 3, 9, 0, 6, 2, 8, 5, 5, 0, 8, 8, 7, 8, 5, 2, 4, 6, 1, 5, 5, 7, 1, 2, 6, 8, 1, 5, 7, 6, 6, 8, 6, 4, 4, 2, 5, 2, 2, 5, 5, 5

Offset: 1

Eric W. Weisstein

An algebraic integer of degree 71. - Charles R Greathouse IV, Aug 10 2014

			1.303577269034296391257099112152551890730702504659404875754861390628550...

John 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.
John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 209.
S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 452-455.
Clifford A. Pickover, The Math Book, From Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics, Sterling Publ., NY, 2009, page 486.
I. Vardi, Computational Recreations in Mathematica. Addison-Wesley, Redwood City, CA, 1991, p. 4.

Edward Krogius Table of n, a(n) for n = 1..50000 (terms 1..20000 from Harry J. Smith).
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.
Jonathan Comes, Stuttering look and say sequences and a challenger to Conway's most complicated algebraic number from the silliest source, arXiv:2206.11991 [math.HO], 2022.
John H. Conway and Brady Haran, Look-and-Say Numbers (2014).
S. R. Finch, Conway's Constant
S. R. Finch, Conway's Constant [From the Wayback Machine]
Thomas Morrill, Look, Knave, arXiv:2004.06414 [math.CO], 2020.
Eric Weisstein's World of Mathematics, Conways Constant.
Eric Weisstein's World of Mathematics, Lookand Say Sequence.
Index entries for algebraic numbers, degree 71.

Cf. A014967, A005150, A006715, A006751.

RealDigits[ NSolve[{0 == Plus @@ ({1, 0, -1, -2, -1, 2, 2, 1, -1, -1, -1, -1, -1, 2, 5, 3, -2, -10, -3, -2, 6, 6, 1, 9, -3, -7, -8, -8, 10, 6, 8, -5, -12, 7, -7, 7, 1, -3, 10, 1, -6, -2, -10, -3, 2, 9, -3, 14, -8, 0, -7, 9, 3, -4, -10, -7, 12, 7, 2, -12, -4, -2, 5, 0, 1, -7, 7, -4, 12, -6, 3, -6} x^Range[71, 0, -1])}, {x}, 105][[-1, -1, -1]]][[1]] (* Ryan Propper, Jul 29 2005 *)

{ allocatemem(932245000); default(realprecision, 20080); x=NULL; r=solve(x=1, 2,\
x^71-x^69-2*x^68-x^67+2*x^66+2*x^65+x^64-x^63-x^62-x^61-x^60\
-x^59+2*x^58+5*x^57+3*x^56-2*x^55-10*x^54-3*x^53-2*x^52+6*x^51\
+6*x^50+x^49+9*x^48-3*x^47-7*x^46-8*x^45-8*x^44+10*x^43+6*x^42\
+8*x^41-5*x^40-12*x^39+7*x^38-7*x^37+7*x^36+x^35-3*x^34+10*x^33\
+x^32-6*x^31-2*x^30-10*x^29-3*x^28+2*x^27+9*x^26-3*x^25+14*x^24\
-8*x^23-7*x^21+9*x^20+3*x^19-4*x^18-10*x^17-7*x^16+12*x^15\
+7*x^14+2*x^13-12*x^12-4*x^11-2*x^10+5*x^9+x^7-7*x^6+7*x^5\
-4*x^4+12*x^3-6*x^2+3*x-6); for (n=1, 20000, d=floor(r); r=(r-d)*10; write("b014715.txt", n, " ", d)); } \\ Harry J. Smith, May 15 2009

P=Pol([1, 0, -1, -2, -1, 2, 2, 1, -1, -1, -1, -1, -1, 2, 5, 3, -2, -10, -3, -2, 6, 6, 1, 9, -3, -7, -8, -8, 10, 6, 8, -5, -12, 7, -7, 7, 1, -3, 10, 1, -6, -2, -10, -3, 2, 9, -3, 14, -8, 0, -7, 9, 3, -4, -10, -7, 12, 7, 2, -12, -4, -2, 5, 0, 1, -7, 7, -4, 12, -6, 3, -6]); polrootsreal(P)[3] \\ Charles R Greathouse IV, Aug 10 2014

More terms from Eric W. Weisstein, Jul 01 2003

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

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

A123132 Describe prime factorization of n (primes in ascending order and with repetition) (method A - initial term is 2).

Original entry on oeis.org

12, 13, 22, 15, 1213, 17, 32, 23, 1215, 111, 2213, 113, 1217, 1315, 42, 117, 1223, 119, 2215, 1317, 12111, 123, 3213, 25, 12113, 33, 2217, 129, 121315, 131, 52, 13111, 12117, 1517, 2223, 137, 12119, 13113, 3215, 141, 121317, 143, 22111, 2315, 12123

Offset: 2

Herman Jamke (hermanjamke(AT)fastmail.fm), Sep 30 2006

Method A = 'frequency' followed by 'digit'-indication. Say 'what you see' in prime factors of n, n>1.

			2 has "one 2" in its prime decomposition, so a(2)=12.
3 has "one 3" in its prime decomposition, so a(3)=13.
4=2*2 has "two 2" in its prime decomposition, so a(4)=22.
5 has "one 5" in its prime decomposition, so a(5)=15.
6=2*3 has "one 2 and one 3" in its prime decomposition, so a(6)=1213.
.....

Paolo P. Lava, Table of n, a(n) for n = 2..10000
A. Frank & P. Jacqueroux, International Contest, 2001.

Cf. A006751, A027746, A063850.

a[n_] := FromDigits@ Flatten@ IntegerDigits[ Reverse /@ FactorInteger@ n]; a/@ Range[2,30] (* Giovanni Resta, Jun 16 2013 *)

for(n=2,25,factn=factor(n); for(i=1,omega(n),print1(factn[i,2],factn[i,1])); print1(","))

a(n) = my(factn=factor(n), sout = ""); for(i=1, omega(n), sout = concat(sout, Str(factn[i, 2])); sout = concat(sout, Str(factn[i, 1]))); eval(sout); \\ Michel Marcus, Jun 29 2017

A022471 Length of n-th term of A022470.

Original entry on oeis.org

1, 2, 4, 4, 6, 10, 12, 14, 22, 26, 30, 44, 56, 70, 98, 130, 162, 216, 292, 358, 470, 628, 792, 1050, 1384, 1788, 2334, 3072, 3974, 5162, 6784, 8786, 11420, 14992, 19484, 25388, 33160, 43262, 56252, 73392, 95798, 124496, 162556, 212048, 275976, 360154

Offset: 1

Clark Kimberling

nonn

a(n) is the length of the n-th term of many sequences generated by methods A and B, including those shown here:
Method A, 1st term ... Method B, 1st term
A001155, 0
A006751, 2 ......... A022470, 2
A006715, 3 ......... A022499, 3
A001140, 4 ......... A022500, 4
A001141, 5 ......... A022501, 5
A001143, 6 ......... A022502, 6
A001145, 7 ......... A022503, 7
A001151, 8 ......... A022504, 8
A001154, 9 ......... A022505, 9
Clark Kimberling, Jun 14 2013

Peter J. C. Moses, Table of n, a(n) for n = 1..3000

Cf. A022470.

a[0] = 2; a[n_] := a[n] = FromDigits[Flatten[{First[#], Length[#]} & /@   Split[IntegerDigits[a[n - 1]]]]]; Map[Length[IntegerDigits[a[#]]] &, Range[0, 40]] (* Peter J. C. Moses, Jun 14 2013 *)
p = {9, -9, 6, -16, 5, 2, 0, -9, -1, -1, 20, 2, 6, -3, -15, -13, 15, 20, 15, -26, -28, 7, 6, 26, -27, -4, 9, -15, 3, 2, 8, 43, 9, -39, -24, -2, -24, 28, 9, 13, 13, -18, -12, -16, 14, 13, 16, 8, -36, 1, -6, -8, 15, 1, 14, 3, -6, -7, -3, 2, -2, 2, 2, 0, -1, -2, -1, 3, 3, -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, 100}], x] (* Peter J. C. Moses, Jun 16 2013 *)

A138485 Say what you see in previous term, from the right, reporting total number for each digit encountered. Initial term is 1.

Original entry on oeis.org

1, 11, 21, 1112, 1231, 211312, 223113, 232122, 421113, 13311214, 14411223, 13223124, 14322123, 23322114, 14213223, 23322114, 14213223, 23322114, 14213223, 23322114, 14213223, 23322114, 14213223, 23322114, 14213223

Offset: 0

Paolo P. Lava and Giorgio Balzarotti, Mar 20 2008

After a while sequence has period 2 -> {23322114,14213223}

			To get the term after 211312, we say: two 2's, three 1's, one 3's, so 223113.

Cf. A063850, A022482, A005150, A005151, A006751, A006715, A006711, A022506-A022513, A138484, A138486-A138493.

cp's OEIS Frontend

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

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A001154 Describe the previous term! (method A - initial term is 9).

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A138484 Say what you see in previous term, from the right, reporting total number for each digit encountered. Initial term is 0.

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A138493 Say what you see in previous term, from the right, reporting total number for each digit encountered. Initial term is 9.

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A014715 Decimal expansion of Conway's constant.

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

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A123132 Describe prime factorization of n (primes in ascending order and with repetition) (method A - initial term is 2).

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