A013948 -id:A013948 - cp's OEIS frontend

A078649 Numbers n such that A000002(n)=A000002(n+1) where A000002 is the Kolakoski sequence.

2, 4, 8, 11, 13, 16, 18, 22, 26, 28, 31, 35, 38, 40, 44, 48, 51, 53, 56, 58, 62, 65, 67, 70, 74, 78, 80, 83, 85, 89, 92, 94, 97, 99, 103, 107, 110, 112, 115, 119, 121, 124, 126, 130, 133, 135, 138, 140, 144, 148, 150, 153, 157, 160, 162, 165, 167, 171, 175, 178, 180

Offset: 1

Benoit Cloitre, Dec 14 2002

Complement sequence of A054353. - Benoit Cloitre, Feb 07 2009
This sequence is the union of A074262 and A074263. - Nathaniel Johnston, May 02 2011
A054354(a(n)-1) = 0. - Reinhard Zumkeller, Aug 03 2013
This is a subsequence of A216345. In particular, this consists of A216345(i) such that A000002(i) = A216345(i+1)-A216345(i) = 2. A013948 is the sequence of all such i. - Danny Rorabaugh, Mar 13 2015

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

Cf. A054353, A074262, A074263, A013948, A216345.

a078649 n = a078649_list !! (n-1)
a078649_list = map (+ 1) $ filter ((== 0) . a054354) [1..]
-- Reinhard Zumkeller, Aug 03 2013

isA078649 := proc(n)
    if A000002(n) = A000002(n+1) then
        true;
    else
        false;
    end if;
end proc:
A078649 := proc(n)
    option remember;
    if n = 1 then
        2;
    else
        for a from procname(n-1)+1 do
            if isA078649(a) then
                return a;
            end if;
        end do:
    end if;
end proc:
seq(A078649(n),n=1..50) ; # R. J. Mathar, Nov 15 2014

a2 = {1, 2, 2}; Do[ a2 = Join[a2, {1+Mod[n-1, 2]}], {n, 3, 80}, {a2[[n]]}]; a3 = Accumulate[a2]; Complement[ Range[ Last[a3]], a3] (* Jean-François Alcover, Jun 18 2013 *)

a(n) is probably asymptotic to 3*n.

a(n) = A216345(A013948(n)). - Danny Rorabaugh, Mar 13 2015

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]

A013947 Positions of 1's in Kolakoski sequence (A000002).

Original entry on oeis.org

1, 4, 5, 7, 10, 13, 14, 16, 17, 20, 22, 23, 25, 28, 29, 31, 32, 34, 37, 40, 41, 43, 46, 48, 49, 51, 52, 55, 58, 59, 61, 64, 67, 68, 70, 71, 73, 76, 78, 79, 82, 85, 86, 88, 91, 94, 95, 97, 98, 101, 103, 104, 106, 109, 112, 113, 115, 116, 118, 121, 122, 124, 125, 128, 130, 131, 133

Offset: 1

Clark Kimberling

Complement: A013948.

Cf. A000002, A022297, A156077.

For n > 1, a(n) = A022297(n)+1.

A332273 Sizes of maximal weakly decreasing subsequences of A000002.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Mar 08 2020

nonn

			The weakly decreasing 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), (2,2,1,1).

The number of runs in the first n terms of A000002 is A156253.
The weakly increasing version is A332875.
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 - 2) + A000002(2*n - 1) for 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).

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).

A306229 Lexicographically earliest sequence of distinct positive terms such that for any n > 0, K(n) <> K(a(n)) (where K denotes the Kolakoski sequence A000002).

Original entry on oeis.org

2, 1, 4, 3, 6, 5, 8, 7, 10, 9, 13, 14, 11, 12, 16, 15, 18, 17, 20, 19, 22, 21, 24, 23, 26, 25, 28, 27, 30, 29, 33, 35, 31, 36, 32, 34, 38, 37, 40, 39, 42, 41, 44, 43, 46, 45, 48, 47, 50, 49, 53, 54, 51, 52, 56, 55, 58, 57, 60, 59, 62, 61, 64, 63, 67, 68, 65

Offset: 1

Rémy Sigrist, Jan 30 2019

nonn

This sequence is a self-inverse permutation of the natural numbers.

			The first terms of the sequence, alongside K and its ordinal transform, are:
  n   a(n)  K(n)  o(n)
  --  ----  ----  ----
   1     2     1     1
   2     1     2     1
   3     4     2     2
   4     3     1     2
   5     6     1     3
   6     5     2     3
   7     8     1     4
   8     7     2     4
   9    10     2     5
  10     9     1     5
  11    13     2     6
  12    14     2     7
  13    11     1     6
  14    12     1     7
  15    16     2     8
  16    15     1     8
  17    18     1     9
  18    17     2     9
  19    20     2    10
  20    19     1    10

Rémy Sigrist, Table of n, a(n) for n = 1..1000
Rémy Sigrist, PARI program for A306229
Index entries for sequences that are permutations of the natural numbers

See A306230 for a similar sequence.

Cf. A000002, A013947, A013948.

PARI
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See Links section.
```

a(A013947(n)) = A013948(n).
a(A013948(n)) = A013947(n).
o(n) = o(a(n)) where o corresponds to the ordinal transform of A000002.

cp's OEIS Frontend

A078649 Numbers n such that A000002(n)=A000002(n+1) where A000002 is the Kolakoski sequence.

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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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A013947 Positions of 1's in Kolakoski sequence (A000002).

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A332273 Sizes of maximal weakly decreasing subsequences of A000002.

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A332875 Sizes of maximal weakly increasing subsequences of A000002.

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A333229 First sums of the Kolakoski sequence A000002.

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A306229 Lexicographically earliest sequence of distinct positive terms such that for any n > 0, K(n) <> K(a(n)) (where K denotes the Kolakoski sequence A000002).

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