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`%I #9 Oct 31 2023 11:01:42
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`%S 1,2,3,5,6,8,9,10,11,13,14,15,17,18,19,21,22,24,25,26,28,29,30,32,33,
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`%T 34,35,37,38,40,41,42,43,45,46,47,49,50,51,53,54,55,56,58,59,61,62,63,
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`%U 65,66,67,69,70,72,73,74,75,77,78,79,81,82,83,85,86,88
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`%N Positions of 1 in A285960; complement of A285961.
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`%C Conjecture: a(n)/n -> 4/3.
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`%C From _Michel Dekking_, Apr 19 2022: (Start)
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`%C Kimberling's conjecture is equivalent to the property that the frequency of 1's in A285960 is equal to 3/4. This follows directly from the corresponding result in A285961.
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`%C But more is true. The first difference sequence (d(n)) = 1,1,2,1,2,1,1,1,2,1,1,2,1,... of (a(n)) is a morphic sequence.
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`%C From the representation of A285960 by the decoration A-> 11, B-> 10, C-> 110, D-> 1, we see that the differences between occurrences of 1's are also given by a decoration:
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`%C A->11, B->2, C->12, D->1.
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`%C This time one finds that (d(n)) can be obtained as a letter to letter image of a morphic sequence fixed point of a morphism mu on {a,b,c,d,e} given by
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`%C mu: a->ab, b->c, c->ade, d->ce, e->ad,
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`%C with the letter-to-letter map
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`%C lambda: a->1, b->1, c->2, d->2, e->1.
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`%C We have (d(n)) = lambda(z), where z is the fixed point z = abcade... of mu.
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`%C (End)
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`%H Clark Kimberling, <a href="/A285962/b285962.txt">Table of n, a(n) for n = 1..10000</a>
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`%e As a word, A285960 = 111011011110111011..., in which 1 is in positions 1,2,3,5,6,8,...
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`%t s = Nest[Flatten[# /. {0 -> {0, 1}, 1 -> {1, 0}}] &, {0}, 8] (* Thue-Morse, A010060 *)
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`%t w = StringJoin[Map[ToString, s]]
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`%t w1 = StringReplace[w, {"01" -> "1"}]
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`%t st = ToCharacterCode[w1] - 48 (* A285960 *)
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`%t Flatten[Position[st, 0]] (* A285961 *)
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`%t Flatten[Position[st, 1]] (* A285962 *)
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`%Y Cf. A010060, A285960, A285961.
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`%K nonn,easy
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`%O 1,2
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`%A _Clark Kimberling_, May 05 2017
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