A269723 - OEIS

%I #33 Jul 20 2022 18:42:44

%S 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,

%T 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,

%U 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

%N 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.

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

%C 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

%H Chai Wah Wu, <a href="/A269723/b269723.txt">Table of n, a(n) for n = 0..16383</a>

%H <a href="/index/Fi#FIXEDPOINTS">Index entries for sequences that are fixed points of mappings</a>

%e The first few A_k are:

%e A_0 = 0,

%e A_1 = 0,0,1,1,

%e A_2 = 0,0,1,1,0,0,1,1,1,1,0,0,1,1,0,0,

%e 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,

%e ...

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

%o (Python)

%o A269723_list = [0]

%o for _ in range(7):

%o     A269723_list += A269723_list + [1-d for d in A269723_list]*2 # _Chai Wah Wu_, Mar 04 2016

%o (Python)

%o A269723_list = [bin(_&0xaaaaa).count('1')%2 for _ in range(16384)] # _Jon Hart_, Aug 09 2016

%Y The Thue-Morse A010060 and A189718 have similar definitions.

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

%K nonn

%O 0

%A _N. J. A. Sloane_, Mar 04 2016. Thanks to _Chai Wah Wu_ for correcting an error in my initial submission.