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A287521 Positions of 0's in A287520.

%I A287521 #24 Oct 02 2019 21:29:09
%S A287521 1,5,9,11,13,18,20,24,25,29,31,36,37,41,45,47,51,52,56,58,63,65,69,70,
%T A287521 73,77,81,83,85,90,91,95,99,101,105,106,109,113,117,119,121,126,128,
%U A287521 132,133,137,139,144,146,150,151,154,158,162,164,166,171,172,176
%N A287521 Positions of 0's in A287520.
%C A287521 a(n) - a(n-1) is in {1, 2, 3, 4, 5} for n >= 1; also, 3n - a(n) is in {0, 1,2} for n >= 1.  The first 20 numbers 3n - a(n) are 2, 1, 0, 1, 2, 0, 1, 0, 2, 1, 2, 0, 2, 1, 0, 1, 0, 2, 1, 2, with 0's in positions given by A189630 (conjectured), 1's in positions given by A287522 (conjectured), and 2's in positions given by A287521 (conjectured). [Corrected by _Michel Dekking_, Sep 30 2019]
%C A287521 From _Michel Dekking_, Oct 02 2019: (Start)
%C A287521 Proof of the conjecture about the positions of 2's in A287520.
%C A287521 The sequence A287520 =: x is, by definition, the fixed point starting with 0 of the morphism sigma given by
%C A287521     sigma(0) = 012, sigma(1) = 102, sigma(2) = 120.
%C A287521 The sequence A189630 is, by definition, the sequence of positions of 1's in A189628, where A189628 =: y is the unique fixed point of the morphism tau given by
%C A287521     tau(0) = 001, tau(1) = 010.
%C A287521 Let pi be the projection map from {0,1,2}* to {0,1}* given by
%C A287521     pi(0) = pi(1) = 0, pi(2) = 1.
%C A287521 Then pi sigma = tau pi on {0,1,2}*, and so pi(x) = y.
%C A287521 It follows directly that the positions of 2's in x are equal to the positions of 1's = pi(2) in y.
%C A287521 (End)
%H A287521 Clark Kimberling, <a href="/A287521/b287521.txt">Table of n, a(n) for n = 1..10000</a>
%t A287521 s = Nest[Flatten[# /. {0->{0, 1, 2}, 1->{1, 0, 2}, 2->{1, 2, 0}}] &, {0}, 9]; (*A287520*)
%t A287521 Flatten[Position[s, 0]]; (* A287521 *)
%t A287521 Flatten[Position[s, 1]]; (* A287522 *)
%t A287521 Flatten[Position[s, 2]]; (* A189630 , conjectured *)
%Y A287521 Cf. A287520, A287522, A189630.
%K A287521 nonn,easy
%O A287521 1,2
%A A287521 _Clark Kimberling_, May 30 2017