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A286048
Positions of 1 in A286046; complement of A286047.
4
1, 3, 5, 8, 9, 11, 14, 15, 17, 19, 21, 24, 25, 27, 29, 32, 33, 35, 37, 40, 41, 43, 46, 47, 49, 51, 54, 55, 57, 59, 62, 63, 65, 67, 69, 72, 73, 75, 78, 79, 81, 83, 85, 88, 89, 91, 93, 96, 97, 99, 101, 104, 105, 107, 109, 111, 114, 115, 117, 120, 121, 123, 125
OFFSET
1,2
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)) = 2,2,3,1,2,3,1,... 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 1's are given by a decoration:
A->22, B->31, C->231, D->2.
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
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->3, d->1, e->3, 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 1 is in positions 1,3,5,8,9,...
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