%I A096270
%S A096270 0,1,0,1,1,0,1,0,1,1,0,1,1,0,1,0,1,1,0,1,0,1,1,0,1,1,0,1,0,1,1,0,1,1,0,
%T A096270 1,0,1,1,0,1,0,1,1,0,1,1,0,1,0,1,1,0,1,0,1,1,0,1,1,0,1,0,1,1,0,1,1,0,1,
%U A096270 0,1,1,0,1,0,1,1,0,1,1,0,1,0,1,1,0,1,1,0,1,0,1,1,0,1,0,1,1,0,1,1,0,1,0
%N A096270 Fixed point of the morphism 0->01, 1->011.
%C A096270 Another version of the Fibonacci word.
%C A096270 (With offset 1) for k>0, a(ceiling(k*phi^2))=0 and a(floor(k*phi^2))=1 
               where phi=(1+sqrt(5))/2 is the Golden ratio - B. Cloitre (benoit7848c(AT)orange.fr), 
               Apr 01 2006
%C A096270 (With offset 1) for n>1 a(A000045(n))=(1-(-1)^n)/2
%C A096270 Equals the Fibonacci word A005614 with an initial zero.
%C A096270 Also the Sturmian word of slope phi (cf. A144595). - N. J. A. Sloane 
               (njas(AT)research.att.com), jan 13 2009
%D A096270 J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 
               2003.
%F A096270 Conjecture: a(n) is given recursively by a(1)=0 and, for n>1, by a(n)=1 
               if n=F(2k+1) and a(n)=a(n-F(2k+1)) otherwise, where F(2k+1) is the 
               largest odd-index Fibonacci number smaller than or equal to n. (This 
               has been confirmed for more than nine million terms.) The odd-index 
               bisection of the Fibonacci numbers (A001519) is {1, 2, 5, 13, 34, 
               89, ...}. So by the conjecture, we would expect that a(30) = a(30-13) 
               = a(17) = a(17-13) = a(4) = a(4-2) = a(2) = 1, which is in fact correct. 
               - John W. Layman (layman(AT)math.vt.edu), Jun 29 2004
%F A096270 (With offset 1) a(n)=-1+floor(n*phi)-floor((n-1)*phi) where phi=(1+sqrt(5))/
               2 so a(n)=-1+A082389(n) - B. Cloitre (benoit7848c(AT)orange.fr), 
               Apr 01 2006
%t A096270 Nest[ Function[l, {Flatten[(l /. {0 -> {0, 1}, 1 -> {0, 1, 1}})]}], {0}, 
               6] (from Robert G. Wilson v Feb 04 2005)
%o A096270 (PARI) a(n)=-1+floor(n*(1+sqrt(5))/2)-floor((n-1)*(1+sqrt(5))/2) [Cloitre]
%Y A096270 Cf. A003849, A096268, A001519. See A005614, A114986 for other versions.
%Y A096270 Sequence in context: A117872 A089809 A165211 this_sequence A159689 A123640 
               A022924
%Y A096270 Adjacent sequences: A096267 A096268 A096269 this_sequence A096271 A096272 
               A096273
%K A096270 nonn,easy
%O A096270 0,1
%A A096270 N. J. A. Sloane (njas(AT)research.att.com), Jun 22 2004
%E A096270 More terms from John W. Layman (layman(AT)math.vt.edu), Jun 29 2004