Search: id:A014578
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%I A014578
%S A014578 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,
%T A014578 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,
%U A014578 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
%N A014578 Binary expansion of Thue constant (or Roth's constant).
%C A014578 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 (benoit7848c(AT)orange.fr), 
               Feb 09 2003
%C A014578 Bill Gosper, Mar 19 2004: 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.
%C A014578 a(0) = 0; then fixed point of the morphism 1->110, 0->111, starting with 
               a(1) = 1 . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Mar 21 2004
%C A014578 Characteristic function of A007417, i.e. a(n) = 1 if n is in A007417 
               and a(n) = 0 otherwise . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), 
               Mar 21 2004
%C A014578 Multiplicative with a(3^e) = (e+1)%2, a(p^e) = 1 otherwise. David W. 
               Wilson (davidwwilson(AT)comcast.net) Jun 10, 2005.
%C A014578 a(A145204(n)) = 0, a(A007417(n)) = 1. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Oct 04 2008]
%H A014578 <a href="Sindx_Ch.html#char_fns">Index entries for characteristic functions</
               a>
%H A014578 Joerg Arndt, <a href="http://www.jjj.de/fxt/#fxtbook">Fxtbook</a>
%H A014578 Michael Gilleland, <a href="selfsimilar.html">Some Self-Similar Integer 
               Sequences</a>
%H A014578 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               ThueSequence.html">Link to a section of The World of Mathematics.</
               a>
%H A014578 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               ThueConstant.html">Thue Constant</a>
%F A014578 a(0)=0; for n>=1, a(n)=sum(k>=0, (-1)^k*(floor(n/3^k)-floor((n-1)/3^k))) 
               - Benoit Cloitre (benoit7848c(AT)orange.fr), Jun 03 2003
%F A014578 a(0)=0, a(3k)=1-a(k); a(3k+1)=a(3k+2)=1. - Benoit Cloitre (benoit7848c(AT)orange.fr), 
               Mar 19 2004
%F A014578 Sum_{k=0..3^n} a(k) = A015518(n+1) = (-1)^n*A014983(n+1). - DELEHAM Philippe 
               (kolotoko(AT)wanadoo.fr), Mar 31 2004
%F A014578 a(n) = 1 - A007949(n) mod 2 for n>0. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Oct 04 2008]
%t A014578 Nest[ Flatten[ # /. {0 -> {1, 1, 1}, 1 -> {1, 1, 0}}] &, {0}, 5] (from 
               Robert G. Wilson v Mar 09 2005)
%o A014578 (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))))
%Y A014578 Cf. Thue-Morse or parity constant A010060.
%Y A014578 Sequence in context: A000494 A022933 A163532 this_sequence A030190 A157658 
               A123506
%Y A014578 Adjacent sequences: A014575 A014576 A014577 this_sequence A014579 A014580 
               A014581
%K A014578 nonn,cons,mult
%O A014578 0,1
%A A014578 Eric Weisstein (eric(AT)weisstein.com)

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