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A284796 Positions of 1's in A284793.

%I A284796 #20 Nov 25 2019 04:59:37
%S A284796 1,4,7,9,12,16,19,22,25,28,31,33,36,40,43,45,48,52,55,58,61,63,66,70,
%T A284796 73,76,79,81,84,88,91,94,97,100,103,105,108,112,115,117,120,124,127,
%U A284796 130,133,136,139,141,144,148,151,153,156,160,163,166,169,171,174
%N A284796 Positions of 1's in A284793.
%C A284796 This sequence and A284794 and A284795 form a partition of the positive integers. Conjecture: for n>=1, we have a(n)-3n+3 in {0,1}, 3n+2-A284795(n) in {0,1,2,3}, and 3n-2-A284795(n) in {0,1}.
%C A284796 A284793 = (1,-1,0,1,0,-1,1,-1,1,-1,0,1,0,-1,0,1,0,-1,1,-1,0,1,0,-1, ... ); thus
%C A284796 A284794 = (2,6,8,10,14,...)
%C A284796 A284795 = (3,5,11,13,15,...)
%C A284796 A284796 = (1,4,7,9,12,15,...).
%C A284796 From _Michel Dekking_, Nov 24 2019: (Start)
%C A284796 Here is a proof of Kimberling's conjecture, i.e., the sequence y defined by y(n) := a(n)-3n+3 takes only values in the alphabet {0,1}.  We know that A284793 = 1,-1,0,1,0,-1,... is a morphic sequence(see A284793). Let tau on the alphabet {A,B,C,D} be given by
%C A284796       A -> BC,  B->BC,  C->ABDC,  D->ABDC.
%C A284796 The unique fixed point of tau is x = BCABDCBC... The letter-to letter map pi which gives A284793 = pi(x) is given by
%C A284796       pi(A)=0,   pi(B)=1,   pi(C)=-1,   pi(D)=0.
%C A284796 The return words of B (i.e., the words with prefix B and no other occurrences of B) in x are
%C A284796       a:= BCA, b:= BDC, c:= BC, d:= BDCA.
%C A284796 The morphism tau induces a so-called derivated morphism on the alphabet of return words, which is given by
%C A284796       beta(a) = abc, beta(b) = adb, beta(c) = ab, beta(d) = adbc.
%C A284796 Since B is the unique letter in {A,B,C,D} projecting on the letter 1, the difference sequence Delta*(a(n)) is given by replacing a,b,c,d by their lengths in the fixed point abcadbab... of beta:
%C A284796       a->3,  b->3,  c->2,  d->4.
%C A284796 The difference sequence (Delta (y(n)) is given by
%C A284796       y(n+1)-y(n) = a(n+1)-a(n)-3.
%C A284796 It follows that Delta y only takes the values 0, -1 and 1. Moreover, the 4 words a,b,c,d  have projections
%C A284796       pi(BCA)=1,-1,0; pi(BDC)=1,0,-1; pi(BC)=1,-1; pi(BDCA)=1,0,-1,0.
%C A284796 From this we see that 1 and -1 always occur in pairs with 1 first, within the 4 projections of a,b,c, and d. Since y(1)=1, this implies that y itself takes only values in {0,1}.
%C A284796 (End)
%H A284796 Clark Kimberling, <a href="/A284796/b284796.txt">Table of n, a(n) for n = 1..10000</a>
%t A284796 s = Nest[Flatten[# /. {0 -> {0, 1}, 1 -> {0, 0, 1, 1}}] &, {0}, 7] (* A284775 *)
%t A284796 d = Differences[s]  (* A284793 *)
%t A284796 Flatten[Position[d, -1]] (* A284794 *)
%t A284796 Flatten[Position[d, 0]]  (* A284795 *)
%t A284796 Flatten[Position[d, 1]]  (* A284796 *)
%t A284796 d1/2  (* positions of 0 in A189664 *)
%Y A284796 Cf. A284793, A284794, A284795, A189664.
%K A284796 nonn,easy
%O A284796 1,2
%A A284796 _Clark Kimberling_, Apr 14 2017