A078649 -id:A078649 - cp's OEIS frontend

A156265 a(n)=3*n-A078649(n).

1, 2, 1, 1, 2, 2, 3, 2, 1, 2, 2, 1, 1, 2, 1, 0, 0, 1, 1, 2, 1, 1, 2, 2, 1, 0, 1, 1, 2, 1, 1, 2, 2, 3, 2, 1, 1, 2, 2, 1, 2, 2, 3, 2, 2, 3, 3, 4, 3, 2, 3, 3, 2, 2, 3, 3, 4, 3, 2, 2, 3, 2, 1, 2, 2, 1, 0, 1, 1, 2, 1, 1, 2, 2, 1, 2, 2, 3, 3, 2, 1, 2, 2, 3, 2, 2, 3, 3, 2, 1, 2, 2, 1, 1, 2, 1, 0, 0, 1, 1, 0, 1, 1, 2, 1

Offset: 1

Benoit Cloitre, Feb 07 2009

nonn

Conjectured to be o(n)

A156267 a(n)=A054353(2*n)-A078649(n).

Original entry on oeis.org

1, 2, 1, 1, 2, 3, 3, 2, 1, 2, 2, 1, 1, 2, 1, -1, -1, 1, 1, 2, 1, 1, 2, 2, 1, -1, 1, 1, 2, 1, 1, 2, 3, 3, 2, 1, 1, 2, 2, 1, 2, 3, 3, 2, 3, 4, 4, 5, 3, 3, 4, 3, 2, 3, 4, 4, 5, 3, 2, 3, 3, 2, 1, 2, 2, 1, -1, 1, 1, 2, 1, 1, 2, 2, 1, 2, 3, 4, 3, 2, 1, 2, 3, 3, 2, 3, 4, 3, 2, 1, 2, 2, 1, 1, 2, 1, -1, -1, 1, 1, -1, 1, 1

Offset: 1

Benoit Cloitre, Feb 07 2009

sign

Conjectured to be o(n)

A156268 a(n)=2*A054353(n)-A078649(n).

Original entry on oeis.org

0, 2, 2, 1, 1, 2, 2, 2, 2, 2, 3, 3, 2, 2, 2, 0, -1, 1, 2, 2, 2, 1, 1, 2, 0, 0, 2, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 2, 3, 1, 1, 2, 2, 2, 3, 3, 4, 4, 2, 2, 2, 1, 1, 2, 2, 3, 5, 3, 1, 2, 2, 2, 2, 2, 3, 3, 1, 1, 2, 2, 0, 1, 1, 2, 2, 2, 3, 3, 2, 2, 2, 2, 3, 5, 3, 2, 4, 3, 3, 3, 3, 4, 4, 3, 3, 3, 1, 0, 2, 3, 1, 3, 2, 2, 2

Offset: 1

Benoit Cloitre, Feb 07 2009

sign

Conjectured to be o(n)

A156271 a(n)=sum(k=1,n,A000002(A078649(k))).

Original entry on oeis.org

2, 3, 5, 7, 8, 9, 11, 12, 14, 15, 16, 18, 20, 21, 23, 24, 25, 27, 29, 30, 32, 34, 35, 36, 38, 39, 41, 43, 44, 46, 48, 49, 50, 52, 53, 55, 57, 58, 59, 61, 62, 63, 65, 66, 67, 69, 71, 72, 74, 75, 77, 79, 80, 81, 83, 85, 86, 88, 89, 90, 92, 93, 95, 96, 97, 99, 100, 102, 104, 105

Offset: 1

Benoit Cloitre, Feb 07 2009

nonn

A156352 a(n)=A078649(A054353(n)-n+1)-A054353(n).

Original entry on oeis.org

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

Offset: 1

Benoit Cloitre, Feb 08 2009

nonn

a(n)=3 iff {A000002(n),A000002(n+1),A000002(n+2)}={2,1,1}

A054353 Partial sums of Kolakoski sequence A000002.

Original entry on oeis.org

1, 3, 5, 6, 7, 9, 10, 12, 14, 15, 17, 19, 20, 21, 23, 24, 25, 27, 29, 30, 32, 33, 34, 36, 37, 39, 41, 42, 43, 45, 46, 47, 49, 50, 52, 54, 55, 57, 59, 60, 61, 63, 64, 66, 68, 69, 71, 72, 73, 75, 76, 77, 79, 81, 82, 84, 86, 87, 88, 90, 91, 93, 95, 96, 98, 100

Offset: 1

N. J. A. Sloane, May 07 2000

Alternate definition: n such that A000002(n) is different from A000002(n+1). - Nathaniel Johnston, May 02 2011

Nathaniel Johnston, Table of n, a(n) for n = 1..10000
O. Bordelles and B. Cloitre, Bounds for the Kolakoski Sequence, J. Integer Sequences, 14 (2011), #11.2.1.
Bertran Steinsky, A Recursive Formula for the Kolakoski Sequence A000002, J. Integer Sequences, Vol. 9 (2006), Article 06.3.7.

Cf. A000002, A074272, A074286, A074288, A078649, A156077.

Cf. A088568 (partial sums of [3 - 2*A000002(n)]).

a054353 n = a054353_list !! (n-1)
a054353_list = scanl1 (+) a000002_list
-- Reinhard Zumkeller, Aug 03 2013

a2 = {1, 2, 2}; Do[ a2 = Join[a2, {1+Mod[n-1, 2]}], {n, 3, 50}, {a2[[n]] } ]; Accumulate[a2] (* Jean-François Alcover, Jun 18 2013 *)

from itertools import accumulate
def alst(nn):
  K = Kolakoski() # using Kolakoski() in A000002
  return list(accumulate(next(K) for i in range(1, nn+1)))
print(alst(66))   # Michael S. Branicky, Jan 12 2021

A000002(a(n)) = (3+(-1)^n)/2; A000002(a(n)+1)=(3-(-1)^n)/2. - Benoit Cloitre, Oct 16 2005

a(n) = n + A074286(n) = 2*n - A156077(n) = A156077(n) + 2*A074286(n). - Jean-Christophe Hervé, Oct 05 2014

A054354 First differences of Kolakoski sequence A000002.

Original entry on oeis.org

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]

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

Original entry on oeis.org

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]

A156243 Bisection of A054353.

Original entry on oeis.org

1, 5, 7, 10, 14, 17, 20, 23, 25, 29, 32, 34, 37, 41, 43, 46, 49, 52, 55, 59, 61, 64, 68, 71, 73, 76, 79, 82, 86, 88, 91, 95, 98, 101, 104, 106, 109, 113, 116, 118, 122, 125, 128, 131, 134, 137, 141, 143, 146, 149, 152, 155, 158, 161, 164, 168, 170, 173, 176, 179, 182

Offset: 1

Benoit Cloitre, Feb 07 2009

nonn

Partial sums of A332273.

a(n) = A054353(2n-1).
A000002(a(n))=1 and A000002(a(n)+1)=2.
n such that A078649(A054353(a(n)-1)-a(n)+2)-A054353(a(n)-1)=2. [From Benoit Cloitre, Feb 08 2009]

cp's OEIS Frontend

A156265 a(n)=3*n-A078649(n).

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A156267 a(n)=A054353(2*n)-A078649(n).

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A156268 a(n)=2*A054353(n)-A078649(n).

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A156271 a(n)=sum(k=1,n,A000002(A078649(k))).

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A156352 a(n)=A078649(A054353(n)-n+1)-A054353(n).

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A054353 Partial sums of Kolakoski sequence A000002.

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

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A156243 Bisection of A054353.

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