A156728 -id:A156728 - cp's OEIS frontend

A054354 First differences of Kolakoski sequence A000002.

1, 0, -1, 0, 1, -1, 1, 0, -1, 1, 0, -1, 0, 1, -1, 0, 1, 0, -1, 1, -1, 0, 1, -1, 1, 0, -1, 0, 1, -1, 0, 1, -1, 1, 0, -1, 1, 0, -1, 0, 1, -1, 1, 0, -1, 1, -1, 0, 1, -1, 0, 1, 0, -1, 1, 0, -1, 0, 1, -1, 1, 0, -1, 1, 0, -1, 0, 1, -1, 0, 1, -1, 1, 0, -1, 1, -1, 0, 1, 0, -1, 1, 0, -1, 0, 1, -1, 1, 0, -1, 1, 0, -1

Offset: 1

N. J. A. Sloane, May 07 2000

sign

The Kolakoski sequence has only 1's and 2's, and is cubefree. Thus, for all n>=1, a(n) is in {-1, 0, 1}, a(n+1) != a(n), and if a(n) = 0, a(n+1) = -a(n-1), while if a(n) != 0, either a(n+1) = 0 and a(n+2) = -a(n) or a(n+1) = -a(n). A further consequence is that the maximum gap between equal values is 4: for all n, there is an integer k, 1Jean-Christophe Hervé, Oct 05 2014
From Daniel Forgues, Jul 07 2015: (Start)
Second differences: {-1, -1, 1, 1, -2, 2, -1, -1, 2, -1, -1, 1, 1, ...}
The sequence of first differences bounces between -1 and 1 with a slope whose absolute value is either 1 or 2. We can compress the information in the second differences into {-1, 1, -2, 2, -1, 2, -1, 1, ...} since the -1 and the 1 come in pairs; which can be compressed further into {1, 1, 2, 2, 1, 2, 1, 1, ...} since the signs alternate, where we only need to know that the initial sign is negative. (End)
This appears to divide the positive integers into three sets, each with density approaching 1/3. Note there are no adjacent equal parts (as mentioned above). - Gus Wiseman, Oct 10 2024

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

Cf. A000002, A054353.
Positions of 0 are A078649.
For Golomb's sequence (A001462) we have A088517.
Positions of -1 are A156242 (descents).
Positions of 1 are A156243 (ascents).
First differences (or second differences of A000002) are A376604.
The Kolakoski sequence (A000002):
- Restrictions: A074264, A100428, A100429, A156263, A156264, A288605.
- Patterns: A013947, A013948, A022292, A074262, A074263.
- Statistics: A074286, A088568, A156077, A156253.
- Transformations: A054354, A156728, A306323, A332273, A332875, A333229.
Cf. A333254.

a054354 n = a054354_list !! (n-1)
a054354_list = zipWith (-) (tail a000002_list) a000002_list
-- Reinhard Zumkeller, Aug 03 2013

a2 = {1, 2, 2}; Do[ a2 = Join[a2, {1 + Mod[n - 1, 2]}], {n, 3, 70}, {a2[[n]]}]; Differences[a2] (* Jean-François Alcover, Jun 18 2013 *)

Abs(a(n)) = (A000002(n)+A000002(n+1)) mod 2. - Benoit Cloitre, Nov 17 2003

A376604 Second differences of the Kolakoski sequence (A000002). First differences of A054354.

Original entry on oeis.org

-1, -1, 1, 1, -2, 2, -1, -1, 2, -1, -1, 1, 1, -2, 1, 1, -1, -1, 2, -2, 1, 1, -2, 2, -1, -1, 1, 1, -2, 1, 1, -2, 2, -1, -1, 2, -1, -1, 1, 1, -2, 2, -1, -1, 2, -2, 1, 1, -2, 1, 1, -1, -1, 2, -1, -1, 1, 1, -2, 2, -1, -1, 2, -1, -1, 1, 1, -2, 1, 1, -2, 2, -1, -1

Offset: 1

Gus Wiseman, Oct 02 2024

sign

Since A000002 has no runs of length 3, this sequence contains no zeros.

The densities appear to approach (1/3, 1/3, 1/6, 1/6).

			The Kolakoski sequence (A000002) is:
  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, ...
with first differences (A054354):
  1, 0, -1, 0, 1, -1, 1, 0, -1, 1, 0, -1, 0, 1, -1, 0, 1, 0, -1, 1, -1, 0, 1, -1, ...
with first differences (A376604):
  -1, -1, 1, 1, -2, 2, -1, -1, 2, -1, -1, 1, 1, -2, 1, 1, -1, -1, 2, -2, 1, 1, -2, ...

A001462 is Golomb's sequence.
A078649 appears to be zeros of the first and third differences.
A288605 gives positions of first appearances of each balance.
A306323 gives a 'broken' version.
A333254 lists run-lengths of differences between consecutive primes.
For the Kolakoski sequence (A000002):
- Restrictions: A074264, A100428, A100429, A156263, A156264.
- Patterns: A013947, A013948, A022292, A074262, A074263, A156242, A156243.
- Statistics: A054353, A074286, A088568, A156077, A156253.
- Transformations: A054354, A156728, A332273, A332875, A333229, A376604.
For second differences: A036263 (prime), A073445 (composite), A376559 (perfect-power), A376562 (non-perfect-power), A376590 (squarefree), A376593 (nonsquarefree), A376596 (prime-power), A376599 (non-prime-power).

kolagrow[q_]:=If[Length[q]<2,Take[{1,2},Length[q]+1],Append[q,Switch[{q[[Length[Split[q]]]],q[[-2]],Last[q]},{1,1,2},1,{1,2,1},2,{2,1,1},2,{2,1,2},2,{2,2,1},1,{2,2,2},1]]]
kol[n_]:=Nest[kolagrow,{1},n-1];
Differences[kol[100],2]

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

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

A157687 a(n) = n - A054353(A156351(n)).

Original entry on oeis.org

0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0

Offset: 1

Benoit Cloitre, Mar 04 2009

nonn

Cf. A156351, A156728, A157684, A157685, A157686

a(n) = 1 iff n is in A078649, a(n) = 0 iff n is in A054353.
a(n) = n-A054353(A157684(n)+A157685(n)).
a(n) = 1 - A156728(n). - Alan Michael Gómez Calderón, Dec 19 2024

A333229 First sums of the Kolakoski sequence A000002.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Mar 18 2020

nonn

Positions of 3's are A054353.
Positions of 2's are A074262.
Positions of 4's are A074263.
The number of runs in the first n terms of A000002 is A156253(n).
Even-indexed terms are A332273 (without the first term).
Odd-indexed terms are A332875.
Cf. A000002, A001462, A013947, A013948, A088568, A156728, 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];
Table[kol[n][[-1]]+kol[n+1][[-1]],{n,30}]

a(n) = A000002(n) + A000002(n + 1).

cp's OEIS Frontend

A054354 First differences of Kolakoski sequence A000002.

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A376604 Second differences of the Kolakoski sequence (A000002). First differences of A054354.

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

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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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A157687 a(n) = n - A054353(A156351(n)).

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A333229 First sums of the Kolakoski sequence A000002.

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