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A286047
Positions of 0 in A286046; complement of A286048.
4
2, 4, 6, 7, 10, 12, 13, 16, 18, 20, 22, 23, 26, 28, 30, 31, 34, 36, 38, 39, 42, 44, 45, 48, 50, 52, 53, 56, 58, 60, 61, 64, 66, 68, 70, 71, 74, 76, 77, 80, 82, 84, 86, 87, 90, 92, 94, 95, 98, 100, 102, 103, 106, 108, 110, 112, 113, 116, 118, 119, 122, 124
OFFSET
1,1
COMMENTS
Conjecture: 2n - a(n) is in {0,1} for n >= 1.
From Michel Dekking, Apr 12 2022: (Start)
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.
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:
A->22, B->13, C->213, D->2.
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
mu: a->ab, b->cd, c->aed, d->f, e->cd, f->aed,
with the letter-to-letter map
lambda: a->2, b->2, c->1, d->3, e->1, f->2.
We have (d(n)) = lambda(z), where z is the fixed point z = abcdae... of mu.
(End)
LINKS
EXAMPLE
As a word, A286046 = 101010011010011010..., in which 0 is in positions 2,4,6,7,10,...
MATHEMATICA
s = Nest[Flatten[# /. {0 -> {0, 1}, 1 -> {1, 0}}] &, {0}, 9] (* Thue-Morse, A010060 *)
w = StringJoin[Map[ToString, s]]
w1 = StringReplace[w, {"011" -> "1"}]
st = ToCharacterCode[w1] - 48 (* A286046 *)
Flatten[Position[st, 0]] (* A286047 *)
Flatten[Position[st, 1]] (* A286048 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, May 07 2017
STATUS
approved