%I
%S 0,0,1,0,0,1,0,0,1,0,0,0,1,0,0,1,0,0,1,0,0,0,1,0,0,1,0,0,1,0,0,0,1,0,
%T 0,1,0,0,1,0,0,1,0,0,0,1,0,0,1,0,0,1,0,0,0,1,0,0,1,0,0,1,0,0,0,1,0,0,
%U 1,0,0,1,0,0,1,0,0,0,1,0,0,1,0,0,1,0,0,0,1,0,0,1,0,0,1,0,0,0,1,0,0,1,0,0,1,0,0
%N Trajectory of 0 under the morphism 0 > 001, 1 > 0010.
%C Or, fixed point of the morphism 0 > (0,0,1), 1 > (0,0,1,0).
%C Each 0 is replaced by the subsequence (0,0,1) and each 1 is replaced by the subsequence (0,0,1,0). It is easily seen that the only possible fixed point must start with 0. From there on the (initial segment of arbitrary length of the) fixed point can be obtained by simply iterating the map starting from this initial value.  _M. F. Hasler_, Oct 03 2016
%C The Beatty sequence for beta := (3 + sqrt(13))/2, A080081, has the property b(n+1)=b(n)+4 if n is already in the sequence, b(n+1) = b(n) + 3 otherwise. Here, every occurrence of "1" leads to an insertion of one more "0" (3 zeros instead of 2 zeros after the "1"). Therefore A080081(n)1 yields the index of the nth "1" in this sequence, i.e., A0800811 is the characteristic sequence of the present sequence.  _M. F. Hasler_, Oct 07 2016
%C Homogeneous Sturmian sequence with slope alpha = (sqrt(13)  3)/2 = 1/beta.  _Michel Dekking_, Feb 15 2019
%H M. F. Hasler, <a href="/A276397/b276397.txt">Table of n, a(n) for n = 0..12969</a>
%H J.P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, p. 284.
%H T. C. Brown, <a href="http://dx.doi.org/10.4153/CMB19910064">A characterization of the quadratic irrationals</a>, Canad. Math. Bull, 1991, 34(1), 3641.
%H <a href="/index/Fi#FIXEDPOINTS">Index entries for sequences that are fixed points of mappings</a>
%F a(n) = floor((n+2)*alpha)  floor((n+1)*alpha), where alpha = (sqrt(13)3)/2.  _Michel Dekking_, Feb 15 2019
%t Nest[ Flatten[ # /. {0 > {0, 0, 1}, 1 > {0, 0, 1, 0}}] &, {1}, 6]
%o (PARI) a=[0,0,1,0];while(#a<10^4,a=concat(t=apply(i>a[1..i+3],a))) \\ _M. F. Hasler_, Oct 03 2016
%Y Different from A125117 and A144597.
%Y Cf. A085550 ((sqrt(13)3)/2).
%K nonn
%O 0
%A _N. J. A. Sloane_, Sep 11 2016
