A074286 Partial sum of the Kolakoski sequence (A000002) minus n.

0, 1, 2, 2, 2, 3, 3, 4, 5, 5, 6, 7, 7, 7, 8, 8, 8, 9, 10, 10, 11, 11, 11, 12, 12, 13, 14, 14, 14, 15, 15, 15, 16, 16, 17, 18, 18, 19, 20, 20, 20, 21, 21, 22, 23, 23, 24, 24, 24, 25, 25, 25, 26, 27, 27, 28, 29, 29, 29, 30, 30, 31, 32, 32, 33, 34, 34, 34, 35, 35

Offset

1,3

Comments

a(n) is the number of 2's in the Kolakoski word of length n (see first formula below). - Jean-Christophe Hervé, Oct 05 2014

Links

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.

Formula

a(n)=#{1<=k<=n : A000002(k)=2}. - Benoit Cloitre, Feb 03 2009

a(n) = A054353(n) - n. - Nathaniel Johnston, May 02 2011

a(n) = n - A156077(n). - Jean-Christophe Hervé, Oct 05 2014

Example

The Kolakoski sequence is 1, 2, 2, 1, 1, 2, 1, 2, 2, 1, 2, 2, 1, 1, 2, 1, 1, 2, 2, 1, 2, ...; the partial sums are 1, 3, 5, 6, 7, 9, ..., so the sequence is 1-1=0, 3-2=1, 5-3=2, 6-4=2, 7-5=2, 9-6=3, ... .

Mathematica

a2 = {1, 2, 2}; Do[ a2 = Join[a2, {1 + Mod[n - 1, 2]}], {n, 3, 50}, {a2[[n]]}]; a3 = Accumulate[a2]; a3 - Range[Length[a3]] (* Jean-François Alcover, Jun 18 2013 *)

Crossrefs

Cf. A000002 (Kolakoski sequence), A054353 (partial sums of K. sequence), A156077 (number of 1's in K. sequence).

Essentially partial sums of A157686.

Sequence in context: A073174 A107631 A029098 * A025769 A103563 A008625

Adjacent sequences: A074283 A074284 A074285 * A074287 A074288 A074289

Keyword

nonn,easy

Author

Jon Perry, Sep 21 2002

Extensions

Corrected offset from Nathaniel Johnston, May 02 2011

Status

approved