A014578 Binary expansion of Thue constant (or Roth's constant).

0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1

Offset

0,1

Comments

a(0)=0; to construct the sequence start with a(1)=1, then concatenate twice and change the last term 1->0 giving 1,1,0. Concatenate those 3 terms twice giving 1,1,0,1,1,0,1,1,0, change the last term 0->1 giving 1,1,0,1,1,0,1,1,1. Concatenate those 9 terms twice and change the last term 1->0, etc. - Benoit Cloitre, Feb 09 2003

It is probably my fault if this constant is misattributed. It was "computed" circa 1971 by a very simple Life pattern (as a diagonal row of blinkers), an obvious case of the (Thue-Siegel-)Roth criterion for transcendence, since the error after 3^n bits is ~2^-3^(n+1) = O(denominator^-3). I probably should have called it Roth's constant. - Bill Gosper, Mar 19 2004

a(0) = 0; then fixed point of the morphism 1->110, 0->111, starting with a(1) = 1. - Philippe Deléham, Mar 21 2004

Characteristic function of A007417, i.e., a(n) = 1 if n is in A007417 and a(n) = 0 otherwise. - Philippe Deléham, Mar 21 2004

Multiplicative with a(3^e) = (e+1)%2, a(p^e) = 1 otherwise. - David W. Wilson, Jun 10 2005

a(A145204(n)) = 0, a(A007417(n)) = 1. - Reinhard Zumkeller, Oct 04 2008

1 if the ternary representation of n has an even number of trailing zeros. - Ralf Stephan, Sep 02 2013

Links

Antti Karttunen, Table of n, a(n) for n = 0..100000

Joerg Arndt, Matters Computational (The Fxtbook), section 38.2, pp.730-731

Michael Gilleland, Some Self-Similar Integer Sequences

Eric Weisstein's World of Mathematics, Thue Sequence

Eric Weisstein's World of Mathematics, Thue Constant

Index entries for characteristic functions

Index entries for sequences that are fixed points of mappings

Formula

a(0)=0; for n>=1, a(n)=sum(k>=0, (-1)^k*(floor(n/3^k)-floor((n-1)/3^k))). - Benoit Cloitre, Jun 03 2003

a(0)=0, a(3k)=1-a(k); a(3k+1)=a(3k+2)=1. - Benoit Cloitre, Mar 19 2004

Sum_{k=0..3^n} a(k) = A015518(n+1) = (-1)^n*A014983(n+1). - Philippe Deléham, Mar 31 2004

a(n) = 1 - A007949(n) mod 2 for n>0. - Reinhard Zumkeller, Oct 04 2008

Let T(x) be the g.f., then T(x) + T(x^3) = x/(1-x). - Joerg Arndt, May 11 2010

Asymptotic mean: lim_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 3/4. - Amiram Eldar, Jul 13 2020

Example

Start: 1
Rules:
  1 --> 110
  0 --> 111
-------------
0:   (#=1)
  1
1:   (#=3)
  110
2:   (#=9)
  110110111
3:   (#=27)
  110110111110110111110110110
4:   (#=81)
  110110111110110111110110110110110111110110111110110110110110111110110111110110111
- Joerg Arndt, Jul 06 2011

Mathematica

Nest[ Flatten[ # /. {0 -> {1, 1, 1}, 1 -> {1, 1, 0}}] &, {0}, 6] (* Robert G. Wilson v, Mar 09 2005 *)

Prog

(PARI) a(n)=if(n<1, 0, sum(k=0, ceil(log(n)/log(3)), (-1)^k*(floor(n/3^k)-floor((n-1)/3^k))));
(PARI) A014578(n) = if(!n, n, valuation(n, 3)%2==0); \\ Ralf Stephan, Sep 02 2013, edited for the term a(0)=0 - Antti Karttunen, May 28 2024

Crossrefs

Cf. Thue-Morse or parity constant A010060.

Cf. A154271.

Sequence in context: A188295 A228039 A163532 * A323153 A288861 A030190

Adjacent sequences: A014575 A014576 A014577 * A014579 A014580 A014581

Keyword

nonn,cons,mult

Author

Eric W. Weisstein

Status

approved