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A248056 Positions of 0,0 in the Thue-Morse sequence (A010060).

%I A248056 #12 Jun 16 2025 08:45:43
%S A248056 6,10,18,24,30,34,40,46,54,58,66,72,78,86,90,96,102,106,114,120,126,
%T A248056 130,136,142,150,154,160,166,170,178,184,190,198,202,210,216,222,226,
%U A248056 232,238,246,250,258,264,270,278,282,288,294,298,306,312,318,326,330
%N A248056 Positions of 0,0 in the Thue-Morse sequence (A010060).
%C A248056 Every positive integer lies in exactly one of these four sequences:  A248056, A091855, A091855, A248057.
%H A248056 Clark Kimberling, <a href="/A248056/b248056.txt">Table of n, a(n) for n = 1..1000</a>
%F A248056 a(n) = 2*A091785(n) for n >= 1.
%F A248056 a(n) = A157970(n) + 1. - _Amiram Eldar_, Jun 16 2025
%e A248056 Thue-Morse sequence:  0,1,1,0,1,0,0,1,1,0,0,1,0,1,1,..., so that a(1) = 6 and a(2) = 10.
%t A248056 z = 400; u = Nest[Flatten[# /. {0 -> {0, 1}, 1 -> {1, 0}}] &, {0}, 9] (* A010060 *)
%t A248056 v = Rest[u]
%t A248056 t1 = Table[If[u[[n]] == 0 && v[[n]] == 0, 1, 0], {n, 1, z}];
%t A248056 t2 = Table[If[u[[n]] == 0 && v[[n]] == 1, 1, 0], {n, 1, z}];
%t A248056 t3 = Table[If[u[[n]] == 1 && v[[n]] == 0, 1, 0], {n, 1, z}];
%t A248056 t4 = Table[If[u[[n]] == 1 && v[[n]] == 1, 1, 0], {n, 1, z}];
%t A248056 Flatten[Position[t1, 1]]  (* A248056 *)
%t A248056 Flatten[Position[t2, 1]]  (* A091855 *)
%t A248056 Flatten[Position[t3, 1]]  (* A091785 *)
%t A248056 Flatten[Position[t4, 1]]  (* A248057 *)
%t A248056 SequencePosition[ThueMorse[Range[400]],{0,0}][[All,2]] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale_, Mar 02 2020 *)
%Y A248056 Cf. A010060, A091855, A091785, A157970, A248057.
%K A248056 nonn,easy
%O A248056 1,1
%A A248056 _Clark Kimberling_, Sep 30 2014