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%I #30 Jul 19 2022 22:58:35
%S 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,
%T 15,15,15,16,16,17,18,18,19,20,20,20,21,21,22,23,23,24,24,24,25,25,25,
%U 26,27,27,28,29,29,29,30,30,31,32,32,33,34,34,34,35,35
%N Partial sum of the Kolakoski sequence (A000002) minus n.
%C 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
%H Nathaniel Johnston, <a href="/A074286/b074286.txt">Table of n, a(n) for n = 1..10000</a>
%H O. Bordelles and B. Cloitre, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Bordelles/bordelles7r.html">Bounds for the Kolakoski Sequence</a>, J. Integer Sequences, 14 (2011), #11.2.1.
%H Bertran Steinsky, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Steinsky/steinsky5.html">A Recursive Formula for the Kolakoski Sequence A000002</a>, J. Integer Sequences, Vol. 9 (2006), Article 06.3.7.
%F a(n)=#{1<=k<=n : A000002(k)=2}. - _Benoit Cloitre_, Feb 03 2009
%F a(n) = A054353(n) - n. - _Nathaniel Johnston_, May 02 2011
%F a(n) = n - A156077(n). - _Jean-Christophe Hervé_, Oct 05 2014
%e 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, ... .
%t 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 *)
%Y Cf. A000002 (Kolakoski sequence), A054353 (partial sums of K. sequence), A156077 (number of 1's in K. sequence).
%Y Essentially partial sums of A157686.
%K nonn,easy
%O 1,3
%A _Jon Perry_, Sep 21 2002
%E Corrected offset from _Nathaniel Johnston_, May 02 2011
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