A179854 Number of 0's (mod 3) in the binary expansion of n.

0, 1, 0, 2, 1, 1, 0, 0, 2, 2, 1, 2, 1, 1, 0, 1, 0, 0, 2, 0, 2, 2, 1, 0, 2, 2, 1, 2, 1, 1, 0, 2, 1, 1, 0, 1, 0, 0, 2, 1, 0, 0, 2, 0, 2, 2, 1, 1, 0, 0, 2, 0, 2, 2, 1, 0, 2, 2, 1, 2, 1, 1, 0, 0, 2, 2, 1, 2, 1, 1, 0, 2, 1, 1, 0, 1, 0, 0, 2, 2, 1, 1, 0, 1, 0, 0, 2, 1, 0, 0, 2, 0, 2, 2, 1, 2, 1, 1, 0, 1, 0, 0, 2, 1, 0, 0, 2, 0, 2, 2, 1, 1, 0, 0, 2, 0, 2, 2, 1, 0

Offset: 1

Author

N. J. A. Sloane, Jan 11 2011

Keywords

• nonn

Comments

A ternary analog of A059448.

Offset is 1 to avoid the ambiguity at n=0.

Inspired by Chapter 1 of Allouche and Shallit.

From Michel Dekking, Sep 30 2020: (Start)

Let tau be the "twisted" 3-symbol length 2 Thue-Morse morphism given by

tau(0) = 10, tau(1) = 21, tau (2) = 02.

The name of tau is in analogy with the comments from A297531. The "ordinary" 3-symbol length 2 Thue-Morse morphism is the morphism mu given by

mu(0) = 01, mu(1) = 12, mu(2) = 20.

The unique fixed point of mu is the sequence A071858 = 01121220...

We have mu^3 = tau^3.

The sequence a = (a(n)) satisfies

a = 0 tau(a).

This follows directly from the recursion formulas

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

(End)

References

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

Crossrefs

Cf. A059448. Related to A071858.

Programs

Maple

s1:=[];
for n from 0 to 200 do
t1:=convert(n,base,2); t2:=subs(1=NULL,t1); s1:=[op(s1),nops(t2) mod 3]; od:
s1;

Formula

a(2n) = a(n) + 1 mod 3, a(2n+1) = a(n). - Michel Dekking, Sep 30 2020