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A014578
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Binary expansion of Thue constant (or Roth's constant).
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14
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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
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OFFSET
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0,1
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COMMENTS
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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
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LINKS
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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
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FORMULA
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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
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EXAMPLE
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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
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MATHEMATICA
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Nest[ Flatten[ # /. {0 -> {1, 1, 1}, 1 -> {1, 1, 0}}] &, {0}, 6] (* Robert G. Wilson v, Mar 09 2005 *)
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PROG
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(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 */
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CROSSREFS
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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
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KEYWORD
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nonn,cons,mult
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AUTHOR
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Eric W. Weisstein
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STATUS
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approved
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