A286048 - OEIS

%I #11 Apr 14 2022 15:05:29

%S 1,3,5,8,9,11,14,15,17,19,21,24,25,27,29,32,33,35,37,40,41,43,46,47,

%T 49,51,54,55,57,59,62,63,65,67,69,72,73,75,78,79,81,83,85,88,89,91,93,

%U 96,97,99,101,104,105,107,109,111,114,115,117,120,121,123,125

%N Positions of 1 in A286046; complement of A286047.

%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)) = 2,2,3,1,2,3,1,... of (a(n)) is a morphic sequence.

%C From the representation of A286046 by the decoration A->1010, B->1001, C->101001, D->10, we see that the differences between occurrences of 1's are given by a decoration:

%C       A->22, B->31, C->231, D->2.

%C This is the same decoration as used for A286047, but with the letters 3 and 1 interchanged. It follows directly that (d(n)) can be obtained as a letter to letter image of a morphic sequence, fixed point of 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->3, d->1, e->3, 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="/A286048/b286048.txt">Table of n, a(n) for n = 1..10000</a>

%e As a word, A286046 = 101010011010011010..., in which 1 is in positions 1,3,5,8,9,...

%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, A286047.

%K nonn,easy

%O 1,2

%A _Clark Kimberling_, May 07 2017