A248057 Positions of 1,1 in the Thue-Morse sequence (A010060).

2, 8, 14, 22, 26, 32, 38, 42, 50, 56, 62, 70, 74, 82, 88, 94, 98, 104, 110, 118, 122, 128, 134, 138, 146, 152, 158, 162, 168, 174, 182, 186, 194, 200, 206, 214, 218, 224, 230, 234, 242, 248, 254, 262, 266, 274, 280, 286, 290, 296, 302, 310, 314, 322, 328

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) = 2 and a(2) = 8.
		

Links

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

Crossrefs

Cf. A010060, A091855, A091785, A157971, A248056.

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]],{1,1}][[All,2]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, May 16 2017 *)

PARI

t(n)=hammingweight(n)%2;
for(n=1,500,if(t(n)==1&&t(n-1)==1,print1(n,", "))); \\ Joerg Arndt, Mar 12 2022

Formula

a(n) = 2*A091855(n) for n >= 1.

a(n) = A157971(n) + 1. - Amiram Eldar, Jun 16 2025