A054353 -id:A054353 - 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]

A216345 Position of the beginning of the n-th run in A000002.

Original entry on oeis.org

1, 2, 4, 6, 7, 8, 10, 11, 13, 15, 16, 18, 20, 21, 22, 24, 25, 26, 28, 30, 31, 33, 34, 35, 37, 38, 40, 42, 43, 44, 46, 47, 48, 50, 51, 53, 55, 56, 58, 60, 61, 62, 64, 65, 67, 69, 70, 72, 73, 74, 76, 77, 78, 80, 82, 83, 85, 87, 88, 89, 91, 92, 94, 96, 97, 99, 101, 102, 103

Offset: 1

Jon Perry, Sep 04 2012

nonn

If n is odd then the run is 1's, otherwise it's 2's.

			Kolakoski sequence starts 1221121 so we can see that this sequence begins 1,2,4,6,7.

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

Cf. A000002, A054354.

Cf. A100428, A100429.

a216345 n = a216345_list !! (n-1)
a216345_list = 1 : (filter (\x -> a000002 x /= a000002 (x - 1)) [2..])
-- Reinhard Zumkeller, Aug 03 2013

a=new Array();
a[1]=1;  a[2]=2;  a[3]=2;
cd=1;  ap=3;
for (i=4; i<50; i++)
{
    if (a[ap]==1) a[i]=cd;
    else {a[i]=cd; a[i+1]=cd; i++}
    ap++;
    cd=3-cd;
}
/* document.write(a); document.write("
"); */
b=new Array();
b[1]=1;
c=2;
for (i=2; i<50; i++)
    if (a[i]!=a[i-1]) b[c++]=i;
document.write(b);

{n: A156728(n-1)=1}. - R. J. Mathar, Sep 14 2012

a(n+1) = A054353(n)+1 = a(n) + A000002(n). - Jean-Christophe Hervé, Oct 05 2014

A001083 Length of one version of Kolakoski sequence {A000002(i)} at n-th growth stage.

Original entry on oeis.org

1, 2, 2, 3, 5, 7, 10, 15, 23, 34, 50, 75, 113, 170, 255, 382, 574, 863, 1293, 1937, 2903, 4353, 6526, 9789, 14688, 22029, 33051, 49577, 74379, 111580, 167388, 251090, 376631, 564932, 847376, 1271059, 1906628, 2859984

Offset: 1

N. J. A. Sloane, Simon Plouffe

nonn

			/* generate sequence of sequences by recursion using next1() ( origin 1 ) */
v=[2]; for(n=1,8,p1(v); print1(" -> "); v=next1(v))
2 -> 11 -> 12 -> 122 -> 12211 -> 1221121 -> 1221121221 -> 122112122122112 ->
v=[2]; for(n=1,8,print1(length(v)); print1(","); v=next1(v)) gives: 1,2,2,3,5,7,10,15,

Konstantinos Lambropoulos and Constantinos Simserides, Spectral, localization and charge transport properties of periodic, aperiodic and random binary sequences, arXiv:1808.04764 [cond-mat.soft], 2018.
Eric Weisstein's World of Mathematics, Kolakoski Sequence

Cf. A000002, A042942, A054353.

/* generate sequence starting at 1 given run length sequence */
next1(v)=local(w); w=[]; for(n=1,length(v), for(i=1,v[n],w=concat(w,2-n%2))); w
/* print a number or sequence recursively with no commas */
p1(v)=if(type(v)!="t_VEC",print1(v), for(n=1,length(v),p1(v[n])))

Conjecture: a(n) is asymptotic to c*(3/2)^n where c=0.5819.... - Benoit Cloitre, Jun 01 2004
For n >= 1, a(n+3) = S^n(2) where S(n) = A054353(n) and S^k(2) = S(S^(k-1)(2)). - Benoit Cloitre, Feb 24 2009 [adjusted to match sequence offset by Jon Maiga, Jul 27 2022]
Equivalently, a(n) = A054353(a(n-1)) for n>3. - Jon Maiga, Jul 10 2022

Corrected by and better description from Michael Somos, May 05 2000

A074272 Partial alternating sums of the Kolakoski sequence A000002.

Original entry on oeis.org

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

Offset: 1

Jon Perry, Sep 21 2002

			Kolakoski: 1,2,2,1,1,2,1,2,2,1,2,2,1,1,... We look at 1-2+2-1+1-2+1-2+2 and this gives 1,-1,1,0,1,-1,0,-2,0

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

Cf. A054353.

A249942 Ranks of single 1's in the Kolakoski sequence A000002.

Original entry on oeis.org

7, 10, 20, 25, 34, 37, 43, 46, 55, 61, 64, 73, 76, 82, 88, 91, 101, 106, 109, 118, 128, 137, 143, 146, 152, 155, 164, 170, 173, 182, 187, 196, 199, 205, 211, 214, 223, 233, 236, 241, 251, 260, 263, 268, 277, 280, 286, 289, 298, 301, 307, 313, 316, 326, 331, 334

Offset: 1

Jean-Christophe Hervé, Nov 08 2014

nonn

The single 1's whose ranks are given by this sequence are the 1's between two 2's in the OK sequence A000002. They are associated with iterated words of the OK sequence that develop themselves around each single 1 in two branches (for a description of the iterated words, see comments in A249507, which gives their lengths). The first term of A000002, which is indeed a single 1 but not between two 2's, is thus not considered here.
Each such single 1 is generated by a preceding 1 in the OK sequence that could be single or double, but each single 1 has a double 1 in its ancestors since the first 1 of the OK sequence has no descendants except itself. The length of the iterated word around a single 1 is linked to the number of generations between itself and its nearest double 1 ancestor (A249949 gives the number of generations of the n-th single 1).
A249948 gives the gaps between single 1's.

Jean-Christophe Hervé, Table of n, a(n) for n = 1..10000

A000002, A054351, A054352, A249093, A249094, A249507, A249508, A249948, A249949.

a = {A054353(2k+1), k>1 and A000002(2k+1) = 1}.
Odd values of A216345: A216345(2k+1), k>0, such that A216345(2k+2) =
A216345(2k+1)+1.

A289323 Number of twos minus number of ones in the first 2^n entries of the Kolakoski sequence, A000002.

Original entry on oeis.org

-1, 0, 0, 0, 0, -2, 0, 0, -2, 0, -2, 0, -6, 6, 0, 6, 44, 26, -20, -48, 52, 58, 104, -82, -250, -270, -474, -1864, -3094, -4588, -2534, -7574, -1522, 1818, 9264, 18082, 8898, -30500, -20586, -3232, -90522, -127446, -231384, -83574, -87364, 267886

Offset: 0

Richard P. Brent, Jul 05 2017

sign

This is equivalent to A289322, since a(n) = (#twos)-(#ones) = 2^n-2*(#ones) in the first 2^n entries of A000002.
For example, a(5)=15-17=(32-17)-17=32-2*17=-2 because there are 15 twos and 17 ones in the first 32=2^5 entries of A000002.
The entries in this sequence appear to be of order 2^(n/2), whereas the entries in A289322 are larger (of order 2^n).

			The first 32 entries of the Kolakoski sequence, A000002, are 12211212212211211221211212211211. From this we see that a(5)=15-17=-2, since among the first 2^5 letters, 15 of them are twos and 17 of them are ones.

See references for A289322.

Richard P. Brent, Table of n, a(n) for n = 0..64

Cf. A000002, A054353, A074286, A088568, A156077, A289322.

a(n) = 2^n - 2*A289322(n) = -A088568(2^n) = 2*A054353(2^n) - 3*2^n = 2^n - 2*A156077(2^n).

A074288 n-th term of the Kolakoski sequence (A000002) multiplied by the n-th partial sum.

Original entry on oeis.org

1, 6, 10, 6, 7, 18, 10, 24, 28, 15, 34, 38, 20, 21, 46, 24, 25, 54, 58, 30, 64, 33, 34, 72, 37, 78, 82, 42, 43, 90, 46, 47, 98, 50, 104, 108, 55, 114, 118, 60, 61, 126, 64, 132, 136, 69, 142, 72, 73, 150, 76, 77, 158, 162, 82, 168, 172, 87, 88, 180, 91, 186

Offset: 1

Jon Perry, Sep 21 2002

			The Kolakoski sequence begins 1,2,2,1,1,2,...  and its sequence of partial sums (A054353) begins 1,3,5,6,7,9,... Therefore the opening terms are 1,6,10,6,7,18.

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

Cf. A054353, A074289.

Edited and offset corrected by Nathaniel Johnston, May 02 2011

A156351 a(n) = Sum_{k=1..n} (-1)^K(k+1)*(K(k+1)-K(k)) where K(k) = A000002(k).

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 5, 5, 6, 7, 7, 8, 8, 9, 10, 10, 11, 11, 12, 13, 14, 14, 15, 16, 17, 17, 18, 18, 19, 20, 20, 21, 22, 23, 23, 24, 25, 25, 26, 26, 27, 28, 29, 29, 30, 31, 32, 32, 33, 34, 34, 35, 35, 36, 37, 37, 38, 38, 39, 40, 41, 41, 42, 43, 43, 44, 44, 45, 46, 46, 47, 48, 49, 49

Offset: 1

Benoit Cloitre, Feb 08 2009

nonn

a(n)=1 plus the number of symbol changes in the first n terms of A078880. - Jean-Marc Fedou and Gabriele Fici, Mar 18 2010

Jon Maiga, Table of n, a(n) for n = 1..10000
J.M. Fedou and G. Fici, Some remarks on differentiable sequences and recursivity, Journal of Integer Sequences 13(3): Article 10.3.2 (2010). [From Jean-Marc Fedou and _Gabriele Fici_, Mar 18 2010]

Partial sums of A156728.

a2 = {1, 2, 2}; Do[ a2 = Join[a2, {1 + Mod[n - 1, 2]}], {n, 3, 80}, {i, 1, a2[[n]]}]; a[n_] := Sum[(-1)^a2[[k + 1]]*(a2[[k + 1]] - a2[[k]]), {k, 1, n}]; Table[a[n], {n, 1, 80}] (* Jean-François Alcover, Jun 18 2013 *)

n - A054353(a(n)) = 1 if n is in A078649, n - A054353(a(n)) = 0 otherwise. A078649(n + 1 - a(n)) - n takes values among {0,1,2,3}.

a(n) = gcd(a(a(n-1)),2) + a(n-2) (conjectured). - Jon Maiga, Dec 07 2021

A289322 Number of 1s in the first 2^n entries of the Kolakoski sequence, A000002.

Original entry on oeis.org

1, 1, 2, 4, 8, 17, 32, 64, 129, 256, 513, 1024, 2051, 4093, 8192, 16381, 32746, 65523, 131082, 262168, 524262, 1048547, 2097100, 4194345, 8388733, 16777351, 33554669, 67109796, 134219275, 268437750, 536872179

Offset: 0

Richard P. Brent, Jul 05 2017

nonn

			The first 32 entries of the Kolakoski sequence, A000002, are 12211212212211211221211212211211. From this we see that a(5)=17, since among the first 2^5 letters, 17 of them are 1s.

Richard P. Brent, Table of n, a(n) for n = 0..64
Richard P. Brent and Judy-anne H. Osborn, A fast algorithm for the Kolakoski sequence, Dec. 2016
J. Nilsson, A Space Efficient Algorithm for the Calculation of the Digit Distribution in the Kolakoski Sequence, arXiv preprint arXiv:1110.4228 [math.CO], 2011.
M. Rao, Trucs et bidules sur la séquence de Kolakoski, Oct. 2012.

Cf. A000002. Analogous for powers of ten is A195206. Equivalent but with smaller entries is A289323. Closely related are A054353, A074286, A088568, A156077.

a(n) = (2^n + A088568(2^n))/2 = (2^n - A289323(n))/2.

A329355 The binary expansion of a(n) is the second through n-th terms of A000002 - 1.

Original entry on oeis.org

0, 1, 3, 6, 12, 25, 50, 101, 203, 406, 813, 1627, 3254, 6508, 13017, 26034, 52068, 104137, 208275, 416550, 833101, 1666202, 3332404, 6664809, 13329618, 26659237, 53318475, 106636950, 213273900, 426547801, 853095602, 1706191204, 3412382409, 6824764818

Offset: 1

Gus Wiseman, Nov 12 2019

nonn

			a(11) = 813 has binary expansion q = {1, 1, 0, 0, 1, 0, 1, 1, 0, 1}, and q + 1 is {2, 2, 1, 1, 2, 1, 2, 2, 1, 2}, which is the second through 11th terms of A000002.

Replacing "A000002 - 1" with "2 - A000002" gives A329356.
Partial sums of A000002 are A054353.
Initial subsequences of A000002 are A329360.
Cf. A211100, A275692, A288605, A296658, A329315, A329316, A329317, A329327, A329361, 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]:=If[n==0,{},Nest[kolagrow,{1},n-1]];
Table[FromDigits[kol[n]-1,2],{n,30}]

cp's OEIS Frontend

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

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A216345 Position of the beginning of the n-th run in A000002.

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A001083 Length of one version of Kolakoski sequence {A000002(i)} at n-th growth stage.

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A074272 Partial alternating sums of the Kolakoski sequence A000002.

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A249942 Ranks of single 1's in the Kolakoski sequence A000002.

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A289323 Number of twos minus number of ones in the first 2^n entries of the Kolakoski sequence, A000002.

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A074288 n-th term of the Kolakoski sequence (A000002) multiplied by the n-th partial sum.

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A156351 a(n) = Sum_{k=1..n} (-1)^K(k+1)*(K(k+1)-K(k)) where K(k) = A000002(k).

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A289322 Number of 1s in the first 2^n entries of the Kolakoski sequence, A000002.

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