%I #7 Jan 26 2019 20:00:56
%S 4,11,16,19,28,35,44,47,52,59,64,67,76,79,84,91,100,107,112,115,124,
%T 131,140,143,148,155,164,171,176,179,188,191,196,203,208,211,220,227,
%U 236,239,244,251,256,259,268,271,276,283,292,299,304,307,316,319,324
%N Positions of 0,1,0 in the Thue-Morse sequence (A010060).
%C Every positive integer lies in exactly one of these six sequences:
%C A248056 (positions of 0,0,1)
%C A248104 (positions of 0,1,0)
%C A157970 (positions of 1,0,0)
%C A157971 (positions of 0,1,1)
%C A248105 (positions of 1,0,1)
%C A248057 (positions of 1,1,0)
%C The terms of the sequence are the positions of the mean of the positions of the three numbers 0, 1, 0. - _Harvey P. Dale_, Jan 26 2019
%H Clark Kimberling, <a href="/A248104/b248104.txt">Table of n, a(n) for n = 1..1000</a>
%e Thue-Morse sequence: 0,1,1,0,1,0,0,1,1,0,0,1,0,1,1,..., so that a(1) = 4 and a(2) = 11.
%t z = 600; u = Nest[Flatten[# /. {0 -> {0, 1}, 1 -> {1, 0}}] &, {0}, 13]; v = Rest[u]; w = Rest[v]; t1 = Table[If[u[[n]] == 0 && v[[n]] == 0 && w[[n]] == 1, 1, 0], {n, 1, z}];
%t t2 = Table[If[u[[n]] == 0 && v[[n]] == 1 && w[[n]] == 0, 1, 0], {n, 1, z}];
%t t3 = Table[If[u[[n]] == 1 && v[[n]] == 0 && w[[n]] == 0, 1, 0], {n, 1, z}];
%t t4 = Table[If[u[[n]] == 0 && v[[n]] == 1 && w[[n]] == 1, 1, 0], {n, 1, z}];
%t t5 = Table[If[u[[n]] == 1 && v[[n]] == 0 && w[[n]] == 1, 1, 0], {n, 1, z}];
%t t6 = Table[If[u[[n]] == 1 && v[[n]] == 1 && w[[n]] == 0, 1, 0], {n, 1, z}];
%t Flatten[Position[t1, 1]] (* A248056 *)
%t Flatten[Position[t2, 1]] (* A248104 *)
%t Flatten[Position[t3, 1]] (* A157970 *)
%t Flatten[Position[t4, 1]] (* A157971 *)
%t Flatten[Position[t5, 1]] (* A248105 *)
%t Flatten[Position[t6, 1]] (* A248057 *)
%t Mean/@SequencePosition[ThueMorse[Range[400]],{0,1,0}] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale_, Jan 26 2019 *)
%Y Cf. A010060, A248056, A157970, A157971, A248105, A248057.
%K nonn,easy
%O 1,1
%A _Clark Kimberling_, Oct 01 2014