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A014578 Binary expansion of Thue constant (or Roth's constant). 14
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 (list; constant; graph; refs; listen; history; text; internal format)
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

Table of n, a(n) for n=0..98.

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

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) a(n)=valuation(n, 3)%2==0; /* Ralf Stephan, Sep 02 2013 */

CROSSREFS

Cf. Thue-Morse or parity constant A010060.

Sequence in context: A000494 A022933 A163532 * A030190 A157658 A123506

Adjacent sequences:  A014575 A014576 A014577 * A014579 A014580 A014581

KEYWORD

nonn,cons,mult

AUTHOR

Eric W. Weisstein

STATUS

approved

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Last modified March 29 13:11 EDT 2017. Contains 284270 sequences.