cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A096268 Period-doubling sequence (or period-doubling word): fixed point of the morphism 0 -> 01, 1 -> 00.

Original entry on oeis.org

0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0
Offset: 0

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Author

N. J. A. Sloane, Jun 22 2004

Keywords

Comments

Take highest power of 2 dividing n (A007814(n+1)), read modulo 2.
For the scale-invariance properties see Hendriks et al., 2012.
This is the sequence that results from the ternary Thue-Morse sequence (A036577) if all twos in that sequence are replaced by zeros. - Nathan Fox, Mar 12 2013
This sequence can be used to draw the Von Koch snowflake with a suitable walk in the plane. Start from the origin then the n-th step is "turn +Pi/3 if a(n)=0 and turn -2*Pi/3 if a(n)=1" (see link for a plot of the first 200000 steps). - Benoit Cloitre, Nov 10 2013
1 iff the number of trailing zeros in the binary representation of n+1 is odd. - Ralf Stephan, Nov 11 2013
Equivalently, with offset 1, the characteristic function of A036554 and an indicator for the A003159/A036554 classification of positive integers. - Peter Munn, Jun 02 2020

Examples

			Start: 0
Rules:
  0 --> 01
  1 --> 00
-------------
0:   (#=1)
  0
1:   (#=2)
  01
2:   (#=4)
  0100
3:   (#=8)
  01000101
4:   (#=16)
  0100010101000100
5:   (#=32)
  01000101010001000100010101000101
6:   (#=64)
  0100010101000100010001010100010101000101010001000100010101000100
7:   (#=128)
  010001010100010001000101010001010100010101000100010001010100010001000101010...
[_Joerg Arndt_, Jul 06 2011]

References

  • Michel Rigo, Formal Languages, Automata and Numeration Systems, 2 vols., Wiley, 2014. Mentions this sequence - see "List of Sequences" in Vol. 2.

Links

Crossrefs

Not the same as A073059!
Swapping 0 and 1 gives A035263.
Cf. A096269, A096270, A071858, A220466, A036577.
Cf. A056832, A123087 (partial sums).
With offset 1, classification indicator for A003159/A036554.
Also with offset 1: A007814 mod 2 (cf. A096271 for mod 3), A048675 mod 2 (cf. A332813 for mod 3), A059975 mod 2.

Programs

  • Haskell
    a096268 = (subtract 1) . a056832 . (+ 1)
-- Reinhard Zumkeller, Jul 29 2014
  • Magma
    [Valuation(n+1, 2) mod 2: n in [0..100]]; // Vincenzo Librandi, Jul 20 2016
  • Maple
    nmax:=104: for p from 0 to ceil(simplify(log[2](nmax))) do for n from 0 to ceil(nmax/(p+2))+1 do a((2*n+1)*2^p-1) := p mod 2 od: od: seq(a(n), n=0..nmax); # Johannes W. Meijer, Feb 02 2013
# second Maple program:
a:= proc(n) a(n):= `if`(n::even, 0, 1-a((n-1)/2)) end:
seq(a(n), n=0..125);  # Alois P. Heinz, Mar 20 2019
  • Mathematica
    Nest[ Flatten[ # /. {0 -> {1, 0}, 1 -> {0, 0}}] &, {1}, 7] (* Robert G. Wilson v, Mar 05 2005 *)
{{0}}~Join~SubstitutionSystem[{0 -> {0, 1}, 1 -> {0, 0}}, {1}, 6] // Flatten (* Michael De Vlieger, Aug 15 2016 *)
  • PARI
    a(n)=valuation(n+1,2)%2 \\ Ralf Stephan, Nov 11 2013
  • Python
    def A096268(n): return (~(n+1)&n).bit_length()&1 # Chai Wah Wu, Jan 09 2023

Formula

Recurrence: a(2*n) = 0, a(4*n+1) = 1, a(4*n+3) = a(n). - Ralf Stephan, Dec 11 2004
The recurrence may be extended backwards, with a(-1) = 1. - S. I. Ben-Abraham, Apr 01 2013
a(n) = 1 - A035263(n-1). - Reinhard Zumkeller, Aug 16 2006
Dirichlet g.f.: zeta(s)/(1+2^s). - Ralf Stephan, Jun 17 2007
Let T(x) be the g.f., then T(x) + T(x^2) = x^2/(1-x^2). - Joerg Arndt, May 11 2010
Let 2^k||n+1. Then a(n)=1 if k is odd, a(n)=0 if k is even. - Vladimir Shevelev, Aug 25 2010
a(n) = A007814(n+1) mod 2. - Robert G. Wilson v, Jan 18 2012
a((2*n+1)*2^p-1) = p mod 2, p >= 0 and n >= 0. - Johannes W. Meijer, Feb 02 2013
a(n) = A056832(n+1) - 1. - Reinhard Zumkeller, Jul 29 2014
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1/3. = Amiram Eldar, Sep 18 2022

Extensions

Corrected by Jeremy Gardiner, Dec 12 2004
More terms from Robert G. Wilson v, Feb 26 2005

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

Original entry on oeis.org

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

Views

Author

Benoit Cloitre, Jun 09 2002

Keywords

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

Crossrefs

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

Programs

  • 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 *)
  • 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

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

Extensions

Edited by Ralf Stephan, Dec 11 2004

A078734 Start with 1,2, concatenate 2^k previous terms and change last term as follows: 1->2, 2->3, 3->1.

Original entry on oeis.org

1, 2, 1, 3, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 1, 3, 1, 2, 1, 3, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 1, 3, 1, 2, 1, 3, 1, 2, 1, 1, 1
Offset: 1

Views

Author

Benoit Cloitre, Dec 21 2002

Keywords

Examples

			Concatenating the first 2 terms 1,2 -> 1,2,1,2 and changing 2->3 gives the first 4 terms: 1,2,1,3.
Concatenating those first 4 terms ->1,2,1,3,1,2,1,3 and changing 3->1 gives the first 8 terms: 1,2,1,3,1,2,1,1.

Links

Crossrefs

Cf. A056832, A035263.

Programs

  • Magma
    [Valuation(n, 2) mod 3 + 1: n in [1..100]]; // Vincenzo Librandi, Aug 01 2018
  • Mathematica
    a[n_] := Mod[IntegerExponent[n, 2], 3] + 1; Array[a, 100] (* Amiram Eldar, Oct 28 2022 *)
  • PARI
    seq(n)={my(v=[1,2]); while(#v < n, v=concat(v,v); v[#v] = v[#v] % 3 + 1); vector(n, i, v[i])} \\ Andrew Howroyd, Jul 31 2018
  • PARI
    a(n) = valuation(n, 2) % 3 + 1; \\ Andrew Howroyd, Jul 31 2018

Formula

(Sum_{k=1..n} a(k))/n -> 1.57.... [This limit is the asymptotic mean of this sequence, 11/7. - Amiram Eldar, Oct 28 2022]
Multiplicative with a(2^e) = (e mod 3) + 1, a(p^e) = 1 for odd prime p. - Andrew Howroyd, Jul 31 2018
From Antti Karttunen, Dec 07 2021: (Start)
a(n) = 1 + A096271(n-1) = 1 + A010872(A007814(n)). [as above]
a(n) = A001511(A050985(n)).
(End)
Dirichlet g.f.: zeta(s)*(4^s+2^(s+1)+3)/(4^s+2^s+1). - Amiram Eldar, Dec 30 2022
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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