A071858 - OEIS

%I #42 Apr 27 2023 07:05:26

%S 0,1,1,2,1,2,2,0,1,2,2,0,2,0,0,1,1,2,2,0,2,0,0,1,2,0,0,1,0,1,1,2,1,2,

%T 2,0,2,0,0,1,2,0,0,1,0,1,1,2,2,0,0,1,0,1,1,2,0,1,1,2,1,2,2,0,1,2,2,0,

%U 2,0,0,1,2,0,0,1,0,1,1,2,2,0,0,1,0,1,1,2,0,1,1,2,1,2,2,0,2,0,0,1,0,1,1,2,0

%N (Number of 1's in binary expansion of n) mod 3.

%C This is the generalized Thue-Morse sequence t_3 (Allouche and Shallit, p. 335).

%C Ternary sequence which is a fixed point of the morphism 0 -> 01, 1 -> 12, 2 -> 20.

%C Sequence is T^(oo)(0) where T is the operator acting on any word on alphabet {0,1,2} by inserting 1 after 0, 2 after 1 and 0 after 2. For instance T(001)=010112, T(120)=122001. - _Benoit Cloitre_, Mar 02 2009

%D J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003.

%H G. C. Greubel, <a href="/A071858/b071858.txt">Table of n, a(n) for n = 0..10000</a>

%H Jin Chen, Zhixiong Wen, Wen Wu, <a href="https://arxiv.org/abs/1802.03610">On the additive complexity of a Thue-Morse like sequence</a>, arXiv:1802.03610 [math.CO], 2018.

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

%F a(n) = A010872(A000120(n)).

%F Recurrence: a(2*n) = a(n), a(2*n+1) = (a(n)+1) mod 3.

%F a(n) = A000695(n) mod 3. - _John M. Campbell_, Jul 16 2016

%t f[n_] := Mod[ Count[ IntegerDigits[n, 2], 1], 3]; Table[ f[n], {n, 0, 104}] (* Or *)

%t Nest[ Flatten[ # /. {0 -> {0, 1}, 1 -> {1, 2}, 2 -> {2, 0}}] &, {0}, 7] (* _Robert G. Wilson v_ Mar 03 2005, modified May 17 2014 *)

%t Table[Mod[DigitCount[n,2,1],3],{n,0,110}] (* _Harvey P. Dale_, Jul 01 2015 *)

%o (PARI) for(n=1,200,print1(sum(i=1,length(binary(n)), component(binary(n),i))%3,","))

%o (PARI) map(d)=if(d==2,[2,0],if(d==1,[1,2],[0,1]))

%o {m=53;v=[];w=[0];while(v!=w,v=w;w=[];for(n=1,min(m,length(v)),w=concat(w,map(v[n]))));for(n=1,2*m,print1(v[n],","))} \\ _Klaus Brockhaus_, Jun 23 2004

%Y Cf. A000120, A010872.

%Y Cf. A010060, A001285, A010059, A048707, A096271, A100619, A179868.

%Y See A245555 for another version.

%K nonn,easy

%O 0,4

%A _Benoit Cloitre_, Jun 09 2002

%E Edited by _Ralf Stephan_, Dec 11 2004