A054353 -id:A054353 - cp's OEIS frontend

A156253 Least k such that A054353(k) >= n.

1, 2, 2, 3, 3, 4, 5, 6, 6, 7, 8, 8, 9, 9, 10, 11, 11, 12, 12, 13, 14, 15, 15, 16, 17, 18, 18, 19, 19, 20, 21, 21, 22, 23, 24, 24, 25, 26, 26, 27, 27, 28, 29, 30, 30, 31, 32, 33, 33, 34, 35, 35, 36, 36, 37, 38, 38, 39, 39, 40, 41, 42, 42, 43, 44, 44, 45, 45, 46, 47, 47, 48, 49, 50

Offset: 1

Benoit Cloitre, Feb 07 2009

nonn

a(n)=1 plus the number of symbol changes in the first n terms of A000002. - Jean-Marc Fedou and Gabriele Fici, Mar 18 2010
From N. J. A. Sloane, Nov 12 2018: (Start)
This seems to be A001462 rewritten so the run lengths are given by A000002. The companion sequence, A000002 rewritten so the run lengths are given by A001462, is A321020.
Note that Kolakoski's sequence A000002 and Golomb's sequence A001462 have very similar definitions, although the asymptotic behavior of A001462 is well-understood, while that of A000002 is a mystery. The asymptotic behavior of the two hybrids A156253 and A321020 might be worth investigating. (End)
To expand upon N. J. A. Sloane's comments, it's worth noting that Golomb's sequence has a formula from Colin Mallows: g(n) = g(n-g(g(n-1))) + 1, which closely resembles a(n) = a(n-gcd(a(a(n-1)),2)) + 1. - Jon Maiga, May 16 2023

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).
Jon Maiga, A Recurrence Related to the Kolakoski Sequence
N. J. A. Sloane, Coordination Sequences, Planing Numbers, and Other Recent Sequences (II), Experimental Mathematics Seminar, Rutgers University, Jan 31 2019, Part I, Part 2, Slides (Mentions this sequence)

Cf. A000002, A001462, A054353, A156351, A321020.

a2 = {1, 2, 2}; Do[ a2 = Join[a2, {1 + Mod[n - 1, 2]}], {n, 3, 80}, {i, 1, a2[[n]]}]; a3 = Accumulate[a2]; a[1] = 1; a[n_] := a[n] = For[k = a[n - 1], True, k++, If[a3[[k]] >= n, Return[k]]]; Table[a[n], {n, 1, 80}] (* Jean-François Alcover, Jun 18 2013 *)
a[1] = 1;
a[n_]:=a[n]=a[n-GCD[a[a[n - 1]], 2]]+1
Array[a, 100] (* Jon Maiga, May 16 2023 *)

Conjecture: a(n) should be asymptotic to 2n/3.
Length of n-th run of the sequence = A000002(n). - Benoit Cloitre, Feb 19 2009
Conjecture: a(n) = (a(a(n-1)) mod 2) + a(n-2) + 1. - Jon Maiga, Dec 09 2021
a(n) = a(n-gcd(a(a(n-1)), 2)) + 1. - Jon Maiga, May 16 2023

A156242 Bisection of A054353.

Original entry on oeis.org

3, 6, 9, 12, 15, 19, 21, 24, 27, 30, 33, 36, 39, 42, 45, 47, 50, 54, 57, 60, 63, 66, 69, 72, 75, 77, 81, 84, 87, 90, 93, 96, 100, 102, 105, 108, 111, 114, 117, 120, 123, 127, 129, 132, 136, 139, 142, 145, 147, 151, 154, 156, 159, 163, 166, 169, 172, 174, 177, 181

Offset: 1

Benoit Cloitre, Feb 07 2009

nonn

Positions of strict descents in the Kolakoski sequence A000002. Strict ascents are A156243. - Gus Wiseman, Mar 31 2020

Cf. A000002, A156243.
The version for prime gaps is A258026.
Sizes of maximal weakly increasing subsequences of A000002 are A332875.
Cf. A054804, A124766, A124769, A225620, A238343, A332273, A333213, A333215, A333256, A333383.

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];
Join@@Position[Partition[kol[100],2,1],{2,1}] (* Gus Wiseman, Mar 31 2020 *)

a(n) = A054353(2n).

A000002(a(n))=2 and A000002(a(n)+1)=1. - Jon Perry, Sep 04 2012

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]

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

A074289 Values of A000002(n)*A054353(n) that are repeated.

Original entry on oeis.org

6, 10, 24, 34, 46, 64, 72, 82, 114, 118, 132, 142, 186, 196, 200, 222, 240, 268, 290, 298, 302, 316, 326, 344, 358, 370, 414, 436, 454, 460, 496, 518, 524, 600, 622, 640, 650, 658, 672, 696, 720, 750, 764, 782, 792, 846, 864, 878, 886, 890, 896, 914, 918

Offset: 1

Jon Perry, Sep 21 2002

Values that appear in A074288 multiple times.

			The Kolakoski sequence begins 1,2,2,1,1,2,1,... and its sequence of partial sums begins 1,3,5,6,7,9,10,... Multiplying the sequences term-by-term gives 1,6,10,6,7,18,10,... Since 6 and 10 appear more than once, they are both in this sequence.

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

Cf. A054353, A074288.

Extended and edited by Nathaniel Johnston, May 02 2011

A156562 a(n) = (-1)^nSum_{k=1..n} A054353(k)(-1)^k.

Original entry on oeis.org

1, 2, 3, 3, 4, 5, 5, 7, 7, 8, 9, 10, 10, 11, 12, 12, 13, 14, 15, 15, 17, 16, 18, 18, 19, 20, 21, 21, 22, 23, 23, 24, 25, 25, 27, 27, 28, 29, 30, 30, 31, 32, 32, 34, 34, 35, 36, 36, 37, 38, 38, 39, 40, 41, 41, 43, 43, 44, 44, 46, 45, 48, 47, 49, 49, 51, 50, 52

Offset: 1

Benoit Cloitre, Feb 10 2009

nonn

Jinyuan Wang, Table of n, a(n) for n = 1..1000

Cf. A054353, A156563.

A156250 Least k such that A078649(k) >= A054353(n).

Original entry on oeis.org

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

Offset: 1

Benoit Cloitre, Feb 07 2009

nonn

Cf. A054353, A078649.

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

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)

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}

cp's OEIS Frontend

A156253 Least k such that A054353(k) >= n.

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A156242 Bisection of A054353.

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A156243 Bisection of A054353.

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

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A074289 Values of A000002(n)*A054353(n) that are repeated.

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A156562 a(n) = (-1)^n*Sum_{k=1..n} A054353(k)*(-1)^k.

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A156250 Least k such that A078649(k) >= A054353(n).

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Formula

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

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

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

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Formula

A156562 a(n) = (-1)^nSum_{k=1..n} A054353(k)(-1)^k.