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

6, 10, 18, 24, 30, 34, 40, 46, 54, 58, 66, 72, 78, 86, 90, 96, 102, 106, 114, 120, 126, 130, 136, 142, 150, 154, 160, 166, 170, 178, 184, 190, 198, 202, 210, 216, 222, 226, 232, 238, 246, 250, 258, 264, 270, 278, 282, 288, 294, 298, 306, 312, 318, 326, 330

Offset: 1

Author

Clark Kimberling, Sep 30 2014

Keywords

• nonn
• easy

Comments

Every positive integer lies in exactly one of these four sequences: A248056, A091855, A091855, A248057.

Examples

			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.
		

Links

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

Crossrefs

Cf. A010060, A091855, A091785, A157970, A248057.

Programs

Mathematica

z = 400; u = Nest[Flatten[# /. {0 -> {0, 1}, 1 -> {1, 0}}] &, {0}, 9] (* A010060 *)
v = Rest[u]
t1 = Table[If[u[[n]] == 0 && v[[n]] == 0, 1, 0], {n, 1, z}];
t2 = Table[If[u[[n]] == 0 && v[[n]] == 1, 1, 0], {n, 1, z}];
t3 = Table[If[u[[n]] == 1 && v[[n]] == 0, 1, 0], {n, 1, z}];
t4 = Table[If[u[[n]] == 1 && v[[n]] == 1, 1, 0], {n, 1, z}];
Flatten[Position[t1, 1]]  (* A248056 *)
Flatten[Position[t2, 1]]  (* A091855 *)
Flatten[Position[t3, 1]]  (* A091785 *)
Flatten[Position[t4, 1]]  (* A248057 *)
SequencePosition[ThueMorse[Range[400]],{0,0}][[All,2]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Mar 02 2020 *)

Formula

a(n) = 2*A091785(n) for n >= 1.

a(n) = A157970(n) + 1. - Amiram Eldar, Jun 16 2025