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

Author

Eric W. Weisstein

Keywords

• nonn
• cons
• mult

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

Examples

			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
		

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

Crossrefs

Cf. Thue-Morse or parity constant A010060.

Cf. A154271.

Programs

Mathematica

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

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

Python

from sympy import multiplicity
def A014578(n): return multiplicity(3,n)&1^1 if n else 0 # Chai Wah Wu, Jan 28 2025

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