A071858 (Number of 1's in binary expansion of n) mod 3.

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, 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, 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

Offset

0,4

Comments

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

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

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

References

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

Links

G. C. Greubel, Table of n, a(n) for n = 0..10000

Jin Chen, Zhixiong Wen, Wen Wu, On the additive complexity of a Thue-Morse like sequence, arXiv:1802.03610 [math.CO], 2018.

Index entries for sequences that are fixed points of mappings

Formula

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

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

a(n) = A000695(n) mod 3. - John M. Campbell, Jul 16 2016

Mathematica

f[n_] := Mod[ Count[ IntegerDigits[n, 2], 1], 3]; Table[ f[n], {n, 0, 104}] (* Or *)
Nest[ Flatten[ # /. {0 -> {0, 1}, 1 -> {1, 2}, 2 -> {2, 0}}] &, {0}, 7] (* Robert G. Wilson v Mar 03 2005, modified May 17 2014 *)
Table[Mod[DigitCount[n, 2, 1], 3], {n, 0, 110}] (* Harvey P. Dale, Jul 01 2015 *)

Prog

(PARI) for(n=1, 200, print1(sum(i=1, length(binary(n)), component(binary(n), i))%3, ", "))
(PARI) map(d)=if(d==2, [2, 0], if(d==1, [1, 2], [0, 1]))
{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

Crossrefs

Cf. A000120, A010872.

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

See A245555 for another version.

Sequence in context: A038717 A073267 A159981 * A122864 A140084 A243747

Adjacent sequences: A071855 A071856 A071857 * A071859 A071860 A071861

Keyword

nonn,easy

Author

Benoit Cloitre, Jun 09 2002

Extensions

Edited by Ralf Stephan, Dec 11 2004

Status

approved