A000002 -id:A000002 - cp's OEIS frontend

A013948 Positions of 2's in Kolakoski sequence (A000002).

2, 3, 6, 8, 9, 11, 12, 15, 18, 19, 21, 24, 26, 27, 30, 33, 35, 36, 38, 39, 42, 44, 45, 47, 50, 53, 54, 56, 57, 60, 62, 63, 65, 66, 69, 72, 74, 75, 77, 80, 81, 83, 84, 87, 89, 90, 92, 93, 96, 99, 100, 102, 105, 107, 108, 110, 111, 114, 117, 119, 120, 123, 126, 127, 129, 132, 135

Offset: 1

Clark Kimberling

Cf. A000002, A054353, A078649.

Complement: A013947.

Numbers n such that A078649(A054353(n-1)-n+2)-A054353(n-1)=1. [Benoit Cloitre, Feb 08 2009]

Least k such that A054353(k)>=A078649(n). a(n)=A078649(n)-n+1. [Benoit Cloitre, Feb 07 2009]

A187199 Parse the Kolakoski sequence A000002 into distinct phrases 1, 2, 21, 12, 122, 1221, 121, 12212, 11, ...; a(n) = length of n-th phrase.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 3, 5, 2, 3, 3, 4, 6, 3, 6, 4, 2, 5, 4, 6, 4, 3, 5, 7, 4, 5, 7, 5, 4, 6, 6, 5, 5, 6, 6, 4, 8, 5, 5, 9, 10, 7, 9, 8, 7, 5, 8, 11, 5, 6, 6, 7, 8, 7, 6, 6, 7, 7, 7, 9, 4, 4, 8, 8, 10, 5, 7, 8, 7, 9, 8, 12, 6, 10, 6, 8, 6, 10, 7, 9, 9, 8, 7, 8, 7, 9, 8, 7, 8, 8, 9, 9, 10, 8, 10, 9, 8, 11, 5, 6

Offset: 1

N. J. A. Sloane, Mar 06 2011

nonn

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program

Cf. A000002, A007782, A109337, A187180-A187188, A187200.

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A249507 Length of full iterated word (direct branch and reverse branch) of the Kolakoski sequence A000002 initiated at A000002(n). If the reverse branch is longer than the direct branch, the total length is counted negatively: if A000002(n) = 2, a(n) = 0; if A000002(n) = 1, a(n) = (ld+lr-1)*sign(ld-lr) with ld = max { k | A000002(n-i+1) = A000002(i), 0A000002(n-i+1) = A000002(i), 0

Original entry on oeis.org

0, 0, -4, 2, 0, 5, 0, 0, 10, 0, 0, -4, 2, 0, -2, 4, 0, 0, -8, 0, -2, 2, 0, 8, 0, 0, -4, 2, 0, -2, 2, 0, 5, 0, 0, 14, 0, 0, -4, 2, 0, 5, 0, 0, -5, 0, -2, 2, 0, -2, 4, 0, 0, 24, 0, 0, -4, 2, 0, 5, 0, 0, 10, 0, 0, -4, 2, 0, -2, 2, 0, 5, 0, 0, -5, 0, -2, 4, 0, 0, 38, 0, 0, -4, 2, 0, 5

Offset: 2

Jean-Christophe Hervé, Oct 30 2014

sign

The Kolakoski sequence A000002 has a fractal structure that appears in the infinite number of iterations and reverse iterations of itself that it contains. Each iteration develops itself in two branches, a right branch in the direct sense, and a left branch in the reverse sense, e.g., 122-1-221121. This sequence gives the length of the full iterated word with its right and left branches starting at position n (with a length = 0 if A000002(n) = 2 <> A000002(1) = 1).
Each iterated word is generated by a preceding and shorter one, starting with 12 or 21 with the 1 in odd position. 12 for example gives, as long as the image of the initial 1 remains in odd position: 12 > 2-1-221 > 122-1-221121 > ... (21 generates the same succession with words in reverse order). The right branch is thus always formed by a term of A054351 (successive generations of the OK sequence), and the left branch by a term of the same sequence in reverse order.
The lengths of the 2 branches are never equal, because a fully symmetric iterated word would necessitate at the beginning the word 12121, which never appears in the OK sequence (it would need 111 before in the OK sequence to appear), but the 2 branches cannot differ by more than two successive generations, again because a greater difference would imply an impossible word at the beginning, namely 1-1221-1.
The iterated words are initiated by the 1's in the OK sequence, but the 1's in pairs 11 can only lead to very short iterated words: (1)-1-2(1) or (1)-1-221(2), or the same in reverse order, with values in this sequence of +/-2 or +/-4. On the contrary, each of the single 1's (212) in the OK sequence (except the first) is associated with an iterated word of length at least five, and (conjecture) it is likely that there is no bound to the length of iterated words associated with a single 1.
The number of iterated words with the right branch longer than the left seems to be well balanced, so that (conjecture) the limit of partial sums of this sequence could be o(n).

			The OK sequence begins as (highlighting the 10th term): 122112122-1-2211211221211... where the iterated word 122-1-221121 of length 10 can be seen around the 10th term; thus a(10) = 10.

Jean-Christophe Hervé, Table of n, a(n) for n = 2..99990

Cf. A000002, A054351, A054352, A249093, A249094, A249508.

a(n) = (A249093(n) + A249094(n) - min(1, A249093(n))) * sign(A249093(n) - A249094(n)).

A249508 Lengths of complete iterations (direct and reverse branches) of the Kolakoski sequence A000002.

Original entry on oeis.org

2, 4, 5, 8, 10, 14, 17, 24, 28, 38, 45, 60, 70, 92, 107, 141, 163, 214, 248, 324, 375, 489, 565, 736, 850, 1107, 1278, 1663, 1920, 2495, 2880, 3742, 4317, 5612, 6474, 8415, 9710, 12626, 14567, 18941, 21857, 28415, 32789, 42629, 49187, 63953, 73793, 95944, 110710, 143946, 166097

Offset: 1

Jean-Christophe Hervé, Oct 30 2014

nonn

See comments in A249507 for a description of iterated words in the Kolakoski sequence.

			The OK sequence begins as (highlighting an iteration of length 10): 122112-1221221121-1221211.., where iterations of length 2, 4, 5 and 8 can also be seen.

Jean-Christophe Hervé, Table of n, a(n) for n = 1..65

Cf. A000002, A054351, A054352, A249093, A249094, A249507.

a(2n+1) = A054352(n+1) + A054352(n) - 1;

a(2n+2) = A054352(n+2) + A054352(n) - 1.

A332875 Sizes of maximal weakly increasing subsequences of A000002.

Original entry on oeis.org

3, 3, 3, 3, 3, 4, 2, 3, 3, 3, 3, 3, 3, 3, 3, 2, 3, 4, 3, 3, 3, 3, 3, 3, 3, 2, 4, 3, 3, 3, 3, 3, 4, 2, 3, 3, 3, 3, 3, 3, 3, 4, 2, 3, 4, 3, 3, 3, 2, 4, 3, 2, 3, 4, 3, 3, 3, 2, 3, 4, 2, 3, 3, 3, 3, 3, 2, 4, 3, 3, 3, 3, 3, 3, 3, 3, 4, 3, 2, 3, 3, 3, 4, 2, 3, 4, 3

Offset: 1

Gus Wiseman, Mar 08 2020

nonn

			The weakly increasing subsequences begin: (1,2,2), (1,1,2), (1,2,2), (1,2,2), (1,1,2), (1,1,2,2), (1,2), (1,1,2), (1,2,2), (1,1,2), (1,1,2), (1,2,2), (1,2,2).

The number of runs in the first n terms of A000002 is A156253.
The weakly decreasing version is A332273.
Cf. A000002, A001462, A013947, A013948, A088568, A288605, A296658, A329315, A329316, A329317, A329362.

kolagrow[q_]:=If[Length[q]<2,Take[{1,2},Length[q]+1],Append[q,Switch[{q[[Length[Split[q]]]],q[[-2]],Last[q]},{1,1,1},0,{1,1,2},1,{1,2,1},2,{1,2,2},0,{2,1,1},2,{2,1,2},2,{2,2,1},1,{2,2,2},1]]]
kol[n_Integer]:=Nest[kolagrow,{1},n-1];
Length/@Split[kol[40],#1<=#2&]

a(n) = A000002(2*n - 1) + A000002(2*n).

A007782 Number of factors in the infinite word formed by the Kolakoski sequence A000002.

Original entry on oeis.org

1, 2, 4, 6, 10, 14, 18, 26, 34, 42, 50, 62, 78, 94, 110, 126, 142, 162, 186, 218, 250, 282, 314, 346, 378, 410, 446, 486, 534, 590, 654, 718, 782, 846, 910, 974, 1038, 1102, 1166, 1234, 1302, 1378, 1458, 1554, 1658, 1774, 1898, 2026, 2154, 2282, 2410, 2538, 2666

Offset: 0

Patricia Lamas (lamas(AT)math.uqam.ca)

a(n) = number of different substrings of length n found in Kolakoski sequence A000002. It is conjectured that a(n) grows like n^(log(3)/log(3/2)).

			For length 3 only the strings 112, 121, 211, 221, 212, 122 occur, so a(3) = 6. For length 4 only the 10 strings 1121, 1122, 1211, 1212, 1221, 2112, 2121, 2122, 2211, 2212 occur.

M. Dekking: "What is the long range order in the Kolakoski sequence?" in: The Mathematics of Long-Range Aperiodic Order, ed. R. V. Moody, Kluwer, Dordrecht (1997), pp. 115-125.

D. Wilson, Table of n, a(n) for n = 0..100.

Cf. A000002.

nMax = 52; A007782[m_] := A007782[m] = (kolak = {1, 2, 2}; For[n = 3, n <= m, n++, For[k = 1, k <= kolak[[n]], k++, AppendTo[ kolak, 1 + Mod[n - 1, 2]]]]; factors[n_] := Table[ kolak[[k ;; k + n - 1]], {k, 1, Length[kolak] - n + 1}]; Table[ factors[n] // Union // Length, {n, 0, nMax}]); A007782[nMax]; A007782[m = 2*nMax]; While[ A007782[m] != A007782[m/2], m = 2*m]; A007782[m] (* Jean-François Alcover, Jul 24 2013 *)

Additional comments from Michael Baake (mbaake(AT)pion09.tphys.physik.uni-tuebingen.de), Feb 19 2001.

A074292 Dominant digit in successive groups of 3 from the Kolakoski sequence (A000002).

Original entry on oeis.org

2, 1, 2, 2, 1, 1, 2, 1, 2, 1, 1, 2, 2, 1, 2, 1, 1, 2, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 2, 1, 2, 2, 1, 1, 2, 1, 1, 2, 2, 1, 2, 1, 1, 2, 1

Offset: 1

Jon Perry, Sep 21 2002

nonn

This appears to be the same as a sequence studied by Claude Lenormand in a letter dated Nov 17 2003: break up the Kolakoski sequence (A000002) into runs of identical symbols and omit one symbol from each run.
The sequence studied by Claude Lenormand is A156257 and is not equal to this one: see A248805 = A156257 - A074292. Differences between the two sequences are at n = 47, 48, 56, 57, 128, 129, 137, 139, 147, 148, 176, 177,... - Jean-Christophe Hervé, Oct 11 2014
As in the Kolakoski sequence, runs in this sequence are of length 1 or 2, because a run XX implies the repetition of exactly the same 3-group in the Kolakoski sequence: -YXX-YXX- or -XXY-XXY- or -XYX-XYX-, and this is not possible 3 times. However, words of the form YXYXY appear in this sequence, but don't appear in the Kolakoski sequence. - Jean-Christophe Hervé, Oct 12 2014

			Kolakoski begins (1,2,2), (1,1,2), (1,2,2), (1,2,2), so this begins 2,1,2,2.

Nathaniel Johnston, Table of n, a(n) for n = 1..10001
Claude Lenormand, Deux transformations sur les mots, Preprint, 5 pages, Nov 17 2003. Apparently unpublished. This is a scanned copy of the version that the author sent to me in 2003. - _N. J. A. Sloane_, Oct 02 2018

Cf. A000002, A074293, A074295, A156257, A248805, A318921.

A074292 := proc(n)
    A000002(3*n-2)+A000002(3*n-1)+A000002(3*n)-3 ;
end proc:
seq(A074292(n),n=1..50) ; # R. J. Mathar, Nov 15 2014

OK = {1, 2, 2}; Do[OK = Join[OK, {1+Mod[n-1, 2]}], {n, 3, 1000}, {OK[[n]]}]; If[Count[#, 1] > 1, 1, 2]& /@ Partition[OK, 3]  (* Jean-François Alcover, Nov 13 2014 *)

a(n)=A000002(3n-2)+A000002(3n-1)+A000002(3n)-3. - Benoit Cloitre, Nov 15 2003

More terms from Ray Chandler, Nov 16 2003

Offset corrected by Jean-Christophe Hervé, Oct 11 2014

A100428 Bisection of Kolakoski sequence A000002.

Original entry on oeis.org

1, 2, 1, 1, 2, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 1, 2, 2, 1, 2, 1, 1, 2, 2, 1, 1, 2, 1, 2, 1, 1, 2, 2, 1, 2, 1, 1, 2, 2, 1, 2, 2, 1, 2, 2, 1, 2, 1, 1, 2, 1, 1, 2, 2, 1, 2, 1, 1, 2, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 1, 1, 2, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 1, 2, 1

Offset: 1

N. J. A. Sloane, Nov 20 2004

Lengths of runs of ones in A000002. - Reinhard Zumkeller, Aug 03 2013

As in the Kolakoski sequence itself, the length of runs in this sequence is 1 or 2 : a run X,X,X is not possible, since it would imply X,Y,X,Y,X in the Kolakoski sequence, which in turn would imply 1, 1, 1 somewhere before (one Y, one X, one Y) which is not possible. - Jean-Christophe Hervé, Oct 04 2014

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A100429, A216345.

a100428 n = a100428_list !! n
a100428_list = f a000002_list where f (u:_:us) = u : f us
-- Reinhard Zumkeller, Aug 03 2013

More terms from Joshua Zucker, May 12 2006

Offset changed by Reinhard Zumkeller, Aug 03 2013

A100429 Bisection of Kolakoski sequence A000002.

Original entry on oeis.org

2, 1, 2, 2, 1, 2, 1, 1, 2, 1, 1, 2, 2, 1, 2, 1, 1, 2, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 1, 2, 2, 1, 2, 1, 1, 2, 2, 1, 1, 2, 1, 2, 1, 1, 2, 2, 1, 2, 1, 2, 2, 1, 1, 2, 2, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 1, 1, 2, 2, 1, 1, 2, 1, 1, 2, 2, 1, 2, 1, 1, 2, 2, 1, 2, 2, 1, 2

Offset: 1

N. J. A. Sloane, Nov 20 2004

Lengths of runs of twos in A000002. - Reinhard Zumkeller, Aug 03 2013

As in the Kolakoski sequence itself, the length of runs in this sequence is 1 or 2 (see A100428). - Jean-Christophe Hervé, Oct 04 2014

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A100428, A216345.

a100429 n = a100429_list !! n
a100429_list = g a000002_list where g (_:v:vs) = v : g vs
-- Reinhard Zumkeller, Aug 03 2013

More terms from Joshua Zucker, May 12 2006

Offset changed by Reinhard Zumkeller, Aug 03 2013

A074262 Positions of '11' in Kolakoski sequence A000002.

Original entry on oeis.org

4, 13, 16, 22, 28, 31, 40, 48, 51, 58, 67, 70, 78, 85, 94, 97, 103, 112, 115, 121, 124, 130, 133, 140, 148, 157, 160, 167, 175, 178, 184, 190, 193, 201, 208, 217, 220, 226, 229, 238, 244, 247, 253, 256, 265, 271, 274, 283, 291, 294, 303, 310, 319, 322, 328

Offset: 1

Jon Perry, Sep 20 2002

			Kolakoski sequence begins 1,2,2,1,1,2,1,2,2,1,2,2,1,1,... so 4 and 13 are in this sequence.

Nathaniel Johnston, Table of n, a(n) for n = 1..10000

Cf. A074263.

a2 = {1, 2, 2}; Do[ a2 = Join[a2, {1 + Mod[n - 1, 2]}], {n, 3, 300}, {i, 1, a2[[n]]}]; Split[a2] /. {1, 1} -> {0, 1} // Flatten // Position[#, 0]& // Flatten (* Jean-François Alcover, Jun 18 2013 *)

Extended and offset changed by Nathaniel Johnston, May 02 2011

cp's OEIS Frontend

A013948 Positions of 2's in Kolakoski sequence (A000002).

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A187199 Parse the Kolakoski sequence A000002 into distinct phrases 1, 2, 21, 12, 122, 1221, 121, 12212, 11, ...; a(n) = length of n-th phrase.

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PARI

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A249508 Lengths of complete iterations (direct and reverse branches) of the Kolakoski sequence A000002.

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A332875 Sizes of maximal weakly increasing subsequences of A000002.

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Mathematica

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A007782 Number of factors in the infinite word formed by the Kolakoski sequence A000002.

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A074292 Dominant digit in successive groups of 3 from the Kolakoski sequence (A000002).

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Maple

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A100428 Bisection of Kolakoski sequence A000002.

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Haskell

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A100429 Bisection of Kolakoski sequence A000002.

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