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A376604 Second differences of the Kolakoski sequence (A000002). First differences of A054354.

%I A376604 #14 Oct 04 2024 00:23:06
%S A376604 -1,-1,1,1,-2,2,-1,-1,2,-1,-1,1,1,-2,1,1,-1,-1,2,-2,1,1,-2,2,-1,-1,1,
%T A376604 1,-2,1,1,-2,2,-1,-1,2,-1,-1,1,1,-2,2,-1,-1,2,-2,1,1,-2,1,1,-1,-1,2,
%U A376604 -1,-1,1,1,-2,2,-1,-1,2,-1,-1,1,1,-2,1,1,-2,2,-1,-1
%N A376604 Second differences of the Kolakoski sequence (A000002). First differences of A054354.
%C A376604 Since A000002 has no runs of length 3, this sequence contains no zeros.
%C A376604 The densities appear to approach (1/3, 1/3, 1/6, 1/6).
%e A376604 The Kolakoski sequence (A000002) is:
%e A376604   1, 2, 2, 1, 1, 2, 1, 2, 2, 1, 2, 2, 1, 1, 2, 1, 1, 2, 2, 1, 2, 1, 1, 2, 1, 2, 2, ...
%e A376604 with first differences (A054354):
%e A376604   1, 0, -1, 0, 1, -1, 1, 0, -1, 1, 0, -1, 0, 1, -1, 0, 1, 0, -1, 1, -1, 0, 1, -1, ...
%e A376604 with first differences (A376604):
%e A376604   -1, -1, 1, 1, -2, 2, -1, -1, 2, -1, -1, 1, 1, -2, 1, 1, -1, -1, 2, -2, 1, 1, -2, ...
%t A376604 kolagrow[q_]:=If[Length[q]<2,Take[{1,2},Length[q]+1],Append[q,Switch[{q[[Length[Split[q]]]],q[[-2]],Last[q]},{1,1,2},1,{1,2,1},2,{2,1,1},2,{2,1,2},2,{2,2,1},1,{2,2,2},1]]]
%t A376604 kol[n_]:=Nest[kolagrow,{1},n-1];
%t A376604 Differences[kol[100],2]
%Y A376604 A001462 is Golomb's sequence.
%Y A376604 A078649 appears to be zeros of the first and third differences.
%Y A376604 A288605 gives positions of first appearances of each balance.
%Y A376604 A306323 gives a 'broken' version.
%Y A376604 A333254 lists run-lengths of differences between consecutive primes.
%Y A376604 For the Kolakoski sequence (A000002):
%Y A376604 - Restrictions: A074264, A100428, A100429, A156263, A156264.
%Y A376604 - Patterns: A013947, A013948, A022292, A074262, A074263, A156242, A156243.
%Y A376604 - Statistics: A054353, A074286, A088568, A156077, A156253.
%Y A376604 - Transformations: A054354, A156728, A332273, A332875, A333229, A376604.
%Y A376604 For second differences: A036263 (prime), A073445 (composite), A376559 (perfect-power), A376562 (non-perfect-power), A376590 (squarefree), A376593 (nonsquarefree), A376596 (prime-power), A376599 (non-prime-power).
%K A376604 sign
%O A376604 1,5
%A A376604 _Gus Wiseman_, Oct 02 2024