A179868 -id:A179868 - cp's OEIS frontend

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

Benoit Cloitre, Jun 09 2002

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

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

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

Cf. A000120, A010872.
Cf. A010060, A001285, A010059, A048707, A096271, A100619, A179868.
See A245555 for another version.

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 *)

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

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

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

Edited by Ralf Stephan, Dec 11 2004

A332251 a(n) is the real part of f(n) defined by f(0) = 0 and f(n+1) = f(n) + i^A000120(n) (where i denotes the imaginary unit). Sequence A332252 gives imaginary parts.

Original entry on oeis.org

0, 1, 1, 1, 0, 0, -1, -2, -2, -2, -3, -4, -4, -5, -5, -5, -4, -4, -5, -6, -6, -7, -7, -7, -6, -7, -7, -7, -6, -6, -5, -4, -4, -4, -5, -6, -6, -7, -7, -7, -6, -7, -7, -7, -6, -6, -5, -4, -4, -5, -5, -5, -4, -4, -3, -2, -2, -2, -1, 0, 0, 1, 1, 1, 0, 0, -1, -2

Offset: 0

Rémy Sigrist, Feb 08 2020

The representation of {f(n)} corresponds to a Lévy C Curve.

			The first terms, alongside f(n) and A000120(n), are:
  n   a(n)  f(n)    A000120(n)
  --  ----  ------  ----------
   0     0       0           0
   1     1       1           1
   2     1     1+i           1
   3     1   1+2*i           2
   4     0     2*i           1
   5     0     3*i           2
   6    -1  -1+3*i           2
   7    -2  -2+3*i           3
   8    -2  -2+2*i           1
   9    -2  -2+3*i           2
  10    -3  -3+3*i           2
  11    -4  -4+3*i           3
  12    -4  -4+2*i           2
  13    -5  -5+2*i           3
  14    -5    -5+i           3
  15    -5      -5           4
  16    -4      -4           1
From _Kevin Ryde_, Sep 24 2020: (Start)
  n    =   2^9 +   2^8 +     2^5 +     2^2 +     2^1 = 806
  f(n) = 1*b^9 + i*b^8 + i^2*b^5 + i^3*b^2 + i^4*b^1 = 23 + 37*i
so a(806) = 23 and A332252(806) = 37.
(End)

Rémy Sigrist, Table of n, a(n) for n = 0..8192
Michael Beeler, R. William Gosper, and Richard Schroeppel, HAKMEM, MIT Artificial Intelligence Laboratory report AIM-239, February 1972. Item 135, The "C" Curve, by Gosper, page 65. Also HTML transcription.
Robert Ferréol (MathCurve), Courbe du C (ou courbe de Lévy) [in French]
Paul Lévy, Les courbes planes ou gauches et les surfaces composées de parties semblables au tout, Journal de l'École Polytechnique, July 1938 pages 227-247, and continued October 1938 pages 249-292.
Kevin Ryde, Iterations of the Lévy C Curve, section Coordinates.
Rémy Sigrist, Colored representation of f(n) in the complex plane for n = 0..2^20 (where the hue is function of n)
Rémy Sigrist, Representation of f(n) for n=0..32 in the complex plan
Wikipedia, Lévy C Curve
Index entries for sequences related to coordinates of 2D curves

Cf. A332252 (imaginary part), A000120 (segment direction), A179868 (segment direction mod 4).

Cf. A332383 (dragon curve).

{ z=0; for (n=0, 67, print1 (real(z) ", "); z += I^hammingweight(n)) }

a(n) = my(v=binary(n),s=1); for(i=2,#v, if(v[i],v[i]=(s*=I))); real(subst(Pol(v),'x,1+I)); \\ Kevin Ryde, Sep 24 2020

For any k >= 0:
- a(2^(4*k))   =    (-4)^k,
- a(2^(4*k+1)) =    (-4)^k,
- a(2^(4*k+2)) =         0,
- a(2^(4*k+3)) = -2*(-4)^k.
From Kevin Ryde, Sep 24 2020: (Start)
With complex b = 1+i,
f(2*n) = b*f(n) and f(2*n+1) = f(2*n) + i^A000120(2*n), expand and step.
f(2^k + r) = b^k + i*f(r), for 0 <= r < 2^k, replication.
f(n) = Sum_{j=0..t} i^j*b^k[j] where binary n = 2^k[0] + ... + 2^k[t] with descending powers k[0] > ... > k[t] >= 0, so change binary to base b with rotating coefficient i^0, i^1, i^2, ... at each 1-bit.
(End)

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

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A332251 a(n) is the real part of f(n) defined by f(0) = 0 and f(n+1) = f(n) + i^A000120(n) (where i denotes the imaginary unit). Sequence A332252 gives imaginary parts.

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