A079730 Kolakoski variation using (1,2,3,4) starting with 1,2.

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

Offset: 1

Author

Benoit Cloitre, Feb 17 2003

Keywords

• nonn

Comments

a(1)=1 then a(n) is the length of n-th run. This kind of Kolakoski variation using(1,2,3,4,...,m) as m grows reaches the Golomb's sequence A001462.

Examples

			Sequence begins: 1,2,2,3,3,4,4,4,1,1,1,2,2,2,2,3,3,3,3, read it as: (1),(2,2),(3,3),(4,4,4),(1,1,1),(2,2,2,2),(3,3,3,3),... then count the terms in parentheses to get: 1,2,2,3,3,4,4,.. which is the same sequence.
		

Links

• Ivan Neretin, Table of n, a(n) for n = 1..10000
• Ulrich Reitebuch, Henriette-Sophie Lipschütz, and Konrad Polthier, Visualizing the Kolakoski Sequence, Bridges Conf. Proc.; Math., Art, Music, Architecture, Culture (2023) 481-484.

Crossrefs

Cf. A000002.

Programs

Mathematica

seed = {1, 2, 3, 4};
w = {};
i = 1;
Do[
  w = Join[w,
    Array[seed[[Mod[i - 1, Length[seed]] + 1]] &,
     If[i > Length[w], seed, w][[i]]]];
  i++
  , {n, 41}];
w

Formula

Partial sum sequence is expected to be asymptotic to 5/2*n.

Extensions

Corrected by Ivan Neretin, Apr 01 2015