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A285962 Positions of 1 in A285960; complement of A285961. 3
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, 34, 35, 37, 38, 40, 41, 42, 43, 45, 46, 47, 49, 50, 51, 53, 54, 55, 56, 58, 59, 61, 62, 63, 65, 66, 67, 69, 70, 72, 73, 74, 75, 77, 78, 79, 81, 82, 83, 85, 86, 88 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Conjecture: a(n)/n -> 4/3.

REFERENCES

From Michel Dekking, Apr 19 2022: (Start)

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.

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.

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:

      A->11, B->2, C->12, D->1.

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

      mu: a->ab, b->c, c->ade, d->ce, e->ad,

with the letter-to-letter map

      lambda:  a->1, b->1, c->2, d->2, e->1.

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

(End)

LINKS

Clark Kimberling, Table of n, a(n) for n = 1..10000

EXAMPLE

As a word, A285960 = 111011011110111011..., in which 1 is in positions 1,2,3,5,6,8,...

MATHEMATICA

s = Nest[Flatten[# /. {0 -> {0, 1}, 1 -> {1, 0}}] &, {0}, 8] (* Thue-Morse, A010060 *)

w = StringJoin[Map[ToString, s]]

w1 = StringReplace[w, {"01" -> "1"}]

st = ToCharacterCode[w1] - 48 (* A285960  *)

Flatten[Position[st, 0]]  (* A285961 *)

Flatten[Position[st, 1]]  (* A285962 *)

CROSSREFS

Cf. A010060, A285960, A285961.

Sequence in context: A039104 A184861 A183572 * A047450 A039073 A026359

Adjacent sequences:  A285959 A285960 A285961 * A285963 A285964 A285965

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, May 05 2017

STATUS

approved

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Last modified May 28 13:05 EDT 2022. Contains 354115 sequences. (Running on oeis4.)