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A286686 Positions of 0 in A286685; complement of A286687.

%I A286686 #12 Sep 19 2019 11:03:42
%S A286686 1,3,6,7,9,12,13,15,17,20,21,23,26,27,29,31,34,35,37,40,41,43,46,47,
%T A286686 49,51,54,55,57,60,61,63,65,68,69,71,74,75,77,80,81,83,85,88,89,91,94,
%U A286686 95,97,99,102,103,105,108,109,111,113,116,117,119,122,123,125
%N A286686 Positions of 0 in A286685; complement of A286687.
%C A286686 a(n) - a(n-1) is in {1,2,3} for n>=2, and a(n)/n -> 2.  These are also the positions of 1 in the {0->10, 1->01}-transform of the Pell word, A171588.
%C A286686 From _Michel Dekking_, Sep 19 2019: (Start)
%C A286686 Here is a precise description of the sequence of first differences.
%C A286686 Let tau be the map tau: 0->01, 1->10. By definition A286685 equals tau(b), where b is the Pell word. The words of length 2 occurring in b are 00, 01 and 10. These are mapped by tau to
%C A286686       tau(00) = 0101, tau(01) = 0110, tau(10) = 1001.
%C A286686 Each of these three four letter words contains exactly 2 0's, occurring among the first two letters and among the last two letters. It follows from this that the overlapping words of length 2 in the Pell word b induce distances between 0's in tau(b) of 2 for 00, of 3 for 01, and of 1 for 10. But then the difference sequence (a(n+1) - a(n)) = 2, 3, 1, 2, 3, 1, 2, ... is equal to the 1->3, 2->1, 3->2 permuted version of the 3-symbol Pell word A294180. (End)
%H A286686 Clark Kimberling, <a href="/A286686/b286686.txt">Table of n, a(n) for n = 1..10000</a>
%e A286686 As a word, A286685 = 01011001011001010110010110..., in which 0 is in positions 1,3,6,7,9,12,...
%t A286686 s = Nest[Flatten[# /. {0 -> {0, 0, 1}, 1 -> {0}}] &, {0}, 6] (* A171588 *)
%t A286686 w = StringJoin[Map[ToString, s]]
%t A286686 w1 = StringReplace[w, {"0" -> "01", "1"->"10"}]
%t A286686 st = ToCharacterCode[w1] - 48 ; (* A286685 *)
%t A286686 p0 = Flatten[Position[st, 0]];  (* A286686 *)
%t A286686 p1 = Flatten[Position[st, 1]];  (* A286687 *)
%Y A286686 Cf. A171588, A286685, A286687.
%K A286686 nonn,easy
%O A286686 1,2
%A A286686 _Clark Kimberling_, May 13 2017