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

Author

Jon Perry, Sep 21 2002

Keywords

• nonn
• easy

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

Examples

			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, ... .
		

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.

Crossrefs

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

Essentially partial sums of A157686.

Programs

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 *)

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

Extensions

Corrected offset from Nathaniel Johnston, May 02 2011