A088203 -id:A088203 - cp's OEIS frontend

A006751 Describe the previous term! (method A - initial term is 2).

2, 12, 1112, 3112, 132112, 1113122112, 311311222112, 13211321322112, 1113122113121113222112, 31131122211311123113322112, 132113213221133112132123222112, 11131221131211132221232112111312111213322112, 31131122211311123113321112131221123113111231121123222112

Offset: 1

N. J. A. Sloane

Method A = 'frequency' followed by 'digit'-indication.
No digit exceeds 3. If the starting number a(1) is a single-digit number greater than 3 this will remain as the last digit, all the remaining in any term being no greater than 3. - Carmine Suriano, Sep 07 2010
a(n) = value of concatenation of n-th row in A088203. - Reinhard Zumkeller, Aug 09 2012
This is because for all n > 1, a(n) begins with 1 or 3 and ends with 2. - Jean-Christophe Hervé, May 07 2013
a(n+1) - a(n) is divisible by 10^5 for n > 5. - Altug Alkan, Dec 04 2015

			E.g. the term after 3112 is obtained by saying "one 3, two 1's, one 2", which gives 132112.

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

Cf. A001140, A001141, A001143, A001145, A001151, A001154, A001155, A005150, A006715, A045918.

Cf. A088203 (continuous version).

a006751 = foldl1 (\v d -> 10 * v + d) . map toInteger . a088203_row
-- Reinhard Zumkeller, Aug 09 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, 2 ][ [ n ] ]; Table[ FromDigits[ F[ n ] ], {n, 11} ] (* Zerinvary Lajos, Mar 21 2007 *)

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

l=[2]
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))
    y=''
    s=1
print(l) # Indranil Ghosh, Jul 05 2017

a(n+1) = A045918(a(n)). - Reinhard Zumkeller, Aug 09 2012

A225224 A continuous "look-and-say" sequence (without repetition, seed 1,1,1).

Original entry on oeis.org

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

Offset: 1

Jean-Christophe Hervé, May 02 2013

A variant of the Conway's 'look-and-say' sequence A005150, without run cut-off. It describes at each step the preceding numbers taken altogether.
The sequence is better described as starting with three 1's: 1, 1, 1, and then 3, 1, and 1, 3, etc., as seed one creates a singular case: 1, then 1, 1, which can be continued either as 2, 1 (ignoring the aforesaid first 1, cf. A221646), or as 3, 1, considering twice the first one.
Contrary to the original look-and-say, this sequence is not base dependent, because figures or group of figures are not aggregated and read as numbers.
The sequence is determined by pairs. Terms of even ranks are counts while odd ranks are numbers.
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.
Two successive odd ranks cannot be equal, which implies that sequences of length three always begin on even rank and that two such sequences never follow each other.
Applying the look-and-say principle to the sequence itself, it is simply shift three ranks to the left.
With seed 2 (resp. 3), the sequence is A088203 (resp. A088204). These two sequences are shifted one rank left by the look-and-say transform.
With seed 2, the sequence A088203 is the concatenation of A006751 (original look-and-say method by blocks): this is because all blocks begin with 1 or 3 and end with 2 and therefore, there is no possible interaction between blocks after concatenation.

			The sequence starts with: 1, 1, 1
The first group has three 1's: 3, 1
The next group has one 3: 1, 3
The next group has two 1's: 2, 1
The next group has one 3: 1, 3
The next group has one 2: 1, 2
The next group has two 1's: 2, 1, etc.

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

Cf. A005150 (original look-and-say sequence).
Cf. A221646 (a close variant with seed 1).
Cf. A225212 (a variant with nested repetitions).
Cf. A088203 (seed 2), A088204 (seed 3).
Cf. A225330 (look-and-repeat).

/* 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 */
  if (say > 0) { /* not the first time */
    while (see <= say) {
      if (a[see]== dg)  c += 1 ;
      else break ;
      see += 1 ;
      }
    }
  else {
   c = 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]; If[say > 0, While[ see <= say, If[a[see] == dg, c += 1, Break[]]; see += 1], c = 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 *)

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

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).

A088205 Infinite audioactive word that shifts 1 place left under "Look and Say" method A, starting with a(0)=0.

Original entry on oeis.org

0, 1, 0, 1, 1, 1, 0, 3, 1, 1, 0, 1, 3, 2, 1, 1, 0, 1, 1, 1, 3, 1, 2, 2, 1, 1, 0, 3, 1, 1, 3, 1, 1, 2, 2, 2, 1, 1, 0, 1, 3, 2, 1, 1, 3, 2, 1, 3, 2, 2, 1, 1, 0, 1, 1, 1, 3, 1, 2, 2, 1, 1, 3, 1, 2, 1, 1, 1, 3, 2, 2, 2, 1, 1, 0, 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, 0, 1, 3, 2, 1

Offset: 0

Paul D. Hanna, Sep 22 2003

nonn

Positions of zero are given by A088206.

The sequence is obtained by starting with 0 and then successively appending the count (number of repetitions) and the value which is seen, term after term: "one zero, one one, one zero, three ones, one zero, ...". Otherwise said, the sequence is defined by "Start with 0 and append the result of "Look and Say" applied to the sequence". The "shifted left under Look and Say" property is thus an immediate consequence of the definition. A less trivial property is that there cannot be a number larger than 3 in this sequence. - M. F. Hasler, Jul 28 2015

			The initial terms {0,1,0,1,1,1,0,3,1,1,0,...} are shifted 1 place left by saying: "one 0, one 1, one 0, three 1's, one 0, ...", which drops the leading 0 to give {1,0,1,1,1,0,3,1,1,0,...}.

Cf. A088206, A088203, A087282, A005150.

a=List(0);my(i=1);while(#a<200,listput(a,1);listput(a,a[i]);while(a[i++]==a[#a],a[#a-1]++));a \\ M. F. Hasler, Jul 28 2015

A088206 Positions of zero in the infinite audioactive word, A088205, which shifts left under "Look and Say" method A, starting with a(1)=0.

Original entry on oeis.org

1, 3, 7, 11, 17, 27, 39, 53, 75, 101, 131, 175, 231, 301, 399, 529, 691, 907, 1199, 1557, 2027, 2655, 3447, 4497, 5881, 7669, 10003, 13075, 17049, 22211, 28995, 37781, 49201, 64193

Offset: 1

Paul D. Hanna, Sep 22 2003

Empirically the partial sums of A022471. - Sean A. Irvine, Jul 13 2022

			The initial terms of A088205, {0,1,0,1,1,1,0,3,1,1,0,...}, are shift left by saying: "one 0, one 1, one 0, three 1's, one 0, ...", which drops the leading 0 to give {1,0,1,1,1,0,3,1,1,0,...}.

Cf. A088205, A088203, A087282, A005150, A022471.

cp's OEIS Frontend

A006751 Describe the previous term! (method A - initial term is 2).

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A225224 A continuous "look-and-say" sequence (without repetition, seed 1,1,1).

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A034002 A005150 expanded into single digits.

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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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A088205 Infinite audioactive word that shifts 1 place left under "Look and Say" method A, starting with a(0)=0.

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A088206 Positions of zero in the infinite audioactive word, A088205, which shifts left under "Look and Say" method A, starting with a(1)=0.

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