A248105 -id:A248105 - cp's OEIS frontend

A248104 Positions of 0,1,0 in the Thue-Morse sequence (A010060).

4, 11, 16, 19, 28, 35, 44, 47, 52, 59, 64, 67, 76, 79, 84, 91, 100, 107, 112, 115, 124, 131, 140, 143, 148, 155, 164, 171, 176, 179, 188, 191, 196, 203, 208, 211, 220, 227, 236, 239, 244, 251, 256, 259, 268, 271, 276, 283, 292, 299, 304, 307, 316, 319, 324

Offset: 1

Clark Kimberling, Oct 01 2014

Every positive integer lies in exactly one of these six sequences:
A248056 (positions of 0,0,1)
A248104 (positions of 0,1,0)
A157970 (positions of 1,0,0)
A157971 (positions of 0,1,1)
A248105 (positions of 1,0,1)
A248057 (positions of 1,1,0)
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

			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.

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A010060, A248056, A157970, A157971, A248105, A248057.

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}];
t2 = Table[If[u[[n]] == 0 && v[[n]] == 1 && w[[n]] == 0, 1, 0], {n, 1, z}];
t3 = Table[If[u[[n]] == 1 && v[[n]] == 0 && w[[n]] == 0, 1, 0], {n, 1, z}];
t4 = Table[If[u[[n]] == 0 && v[[n]] == 1 && w[[n]] == 1, 1, 0], {n, 1, z}];
t5 = Table[If[u[[n]] == 1 && v[[n]] == 0 && w[[n]] == 1, 1, 0], {n, 1, z}];
t6 = Table[If[u[[n]] == 1 && v[[n]] == 1 && w[[n]] == 0, 1, 0], {n, 1, z}];
Flatten[Position[t1, 1]]  (* A248056 *)
Flatten[Position[t2, 1]]  (* A248104 *)
Flatten[Position[t3, 1]]  (* A157970 *)
Flatten[Position[t4, 1]]  (* A157971 *)
Flatten[Position[t5, 1]]  (* A248105 *)
Flatten[Position[t6, 1]]  (* A248057 *)
Mean/@SequencePosition[ThueMorse[Range[400]],{0,1,0}] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jan 26 2019 *)

A381848 Sequence obtained by replacing 3-term subwords of A010060 by 0,1,2,3,4,5 as described in Comments.

Original entry on oeis.org

2, 5, 4, 1, 3, 0, 2, 5, 3, 0, 1, 4, 2, 5, 4, 1, 3, 0, 1, 4, 2, 5, 3, 0, 2, 5, 4, 1, 3, 0, 2, 5, 3, 0, 1, 4, 2, 5, 3, 0, 2, 5, 4, 1, 3, 0, 1, 4, 2, 5, 4, 1, 3, 0, 2, 5, 3, 0, 1, 4, 2, 5, 4, 1, 3, 0, 1, 4, 2, 5, 3, 0, 2, 5, 4, 1, 3, 0, 1, 4, 2, 5, 4, 1, 3, 0

Offset: 1

Clark Kimberling, May 28 2025

nonn

The six 3-term subwords of A010060 are 0,0,1; 0,1,0; 0,1,1; 1,0,0; 1,0,1; 1,1,0. These are coded as 0,1,2,3,4,5 respectively, and then these numbers replace the corresponding subwords in A010060. The positions of 0,1,2,3,4,5 are given by A248956, A248104, A157971, A157970, A248105, A248057, respectively.

			Starting with A010060 = (0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0,...), the successive 3-term subwords are 0,1,1; 1,1,0; 1,0,1; 0,1,0; 1,0,0 ..., which code as 2,5,4,1,3,... .

Cf. A010060, A383999, A248956, A248104, A157971, A157970, A248105, A248057.

Partition[ThueMorse[Range[0, 200]], 3, 1] /. Thread[{{0, 0, 1}, {0, 1, 0}, {0, 1, 1}, {1, 0, 0}, {1, 0, 1}, {1, 1, 0}} -> {0, 1, 2, 3, 4, 5}]  (* Peter J. C. Moses, May 22 2025 *)

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A248104 Positions of 0,1,0 in the Thue-Morse sequence (A010060).

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