A269723 Start with A_0 = 0, then extend by setting B_k = complement of A_k and A_{k+1} = A_k A_k B_k B_k; sequence is limit of A_k as k -> infinity.

0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1

Offset

0

Comments

Equivalently, trajectory of 0 under the morphism 0 -> 0011, 1 -> 1100.

a(n) is the number of 1's, mod 2, in the 2^{odd} positions of the binary representation of n. - Jon Hart, Aug 09 2016

Links

Chai Wah Wu, Table of n, a(n) for n = 0..16383

Bill Allombert and Alain Lasjaunias, On a family of continued fractions in Q((T^1)) associated to infinite binary words derived from the Thue-Morse sequence, arXiv:2505.20102 [math.NT], 2025. See p. 2.

Index entries for sequences that are fixed points of mappings

Formula

a(n) = A010060(n) XOR A341389(n). - Alan Michael Gómez Calderón, Jun 17 2025

Example

The first few A_k are:
A_0 = 0,
A_1 = 0,0,1,1,
A_2 = 0,0,1,1,0,0,1,1,1,1,0,0,1,1,0,0,
A_3 = 0,0,1,1,0,0,1,1,1,1,0,0,1,1,0,0,0,0,1,1,0,0,1,1,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,0,0,1,1,0,0,1,1,1,1,0,0,1,1,0,0,0,0,1,1,0,0,1,1,
...

Mathematica

Table[Mod[#, 2] &@ Count[Drop[#, {1, Length@ #, 2}], 1] &@ Reverse@ IntegerDigits[n, 2], {n, 120}] (* Michael De Vlieger, Aug 11 2016 *)

Prog

(Python)
A269723_list = [0]
for _ in range(7):
    A269723_list += A269723_list + [1-d for d in A269723_list]*2 # Chai Wah Wu, Mar 04 2016
(Python)
A269723_list = [bin(_&0xaaaaa).count('1')%2 for _ in range(16384)] # Jon Hart, Aug 09 2016

Crossrefs

The Thue-Morse A010060 and A189718 have similar definitions.

A355340 gives a perspective of the relationship to Thue-Morse.

Cf. A341389.

Sequence in context: A285501 A288551 A327174 * A284487 A156660 A155899

Adjacent sequences: A269720 A269721 A269722 * A269724 A269725 A269726

Keyword

nonn

Author

N. J. A. Sloane, Mar 04 2016. Thanks to Chai Wah Wu for correcting an error in my initial submission

Status

approved