A286047 - OEIS

%I #9 Apr 14 2022 15:05:13

%S 2,4,6,7,10,12,13,16,18,20,22,23,26,28,30,31,34,36,38,39,42,44,45,48,

%T 50,52,53,56,58,60,61,64,66,68,70,71,74,76,77,80,82,84,86,87,90,92,94,

%U 95,98,100,102,103,106,108,110,112,113,116,118,119,122,124

%N Positions of 0 in A286046; complement of A286048.

%C Conjecture: 2n - a(n) is in {0,1} for n >= 1.

%C From _Michel Dekking_, Apr 12 2022: (Start)

%C Obviously Kimberling's conjecture is equivalent to the property that A286046 is a concatenation of the two 2-blocks 01 and 10. This can be read off immediately from the {A, B, C, D} composition of A286046 given in the comments of that sequence.

%C But more is true. The first difference sequence (d(n)) = 2,2,1,3,2,1,3,...  of (a(n)) is a morphic sequence. From the representation of A286046 by the decoration A->1010, B->1001, C->101001, D->10, we see that the differences between occurrences of 0's are given by a decoration:

%C       A->22, B->13, C->213, D->2.

%C The 'natural' algorithm to obtain (d(n)) as a letter to letter image of a morphic sequence from this decoration yields (for example) a morphism mu on an alphabet {a,b,c,d,e,f} given by

%C       mu: a->ab, b->cd, c->aed, d->f, e->cd, f->aed,

%C with the letter-to-letter map

%C     lambda:  a->2, b->2, c->1, d->3, e->1, f->2.

%C We have (d(n)) = lambda(z), where z is the fixed point z = abcdae... of mu.

%C (End)

%H Clark Kimberling, <a href="/A286047/b286047.txt">Table of n, a(n) for n = 1..10000</a>

%e As a word, A286046 = 101010011010011010..., in which 0 is in positions 2,4,6,7,10,...

%t s = Nest[Flatten[# /. {0 -> {0, 1}, 1 -> {1, 0}}] &, {0}, 9] (* Thue-Morse, A010060 *)

%t w = StringJoin[Map[ToString, s]]

%t w1 = StringReplace[w, {"011" -> "1"}]

%t st = ToCharacterCode[w1] - 48 (* A286046 *)

%t Flatten[Position[st, 0]]  (* A286047 *)

%t Flatten[Position[st, 1]]  (* A286048 *)

%Y Cf. A010060, A286046, A286048.

%K nonn,easy

%O 1,1

%A _Clark Kimberling_, May 07 2017