A285962 - OEIS

A285962

Positions of 1 in A285960; complement of A285961.

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

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OFFSET

1,2

COMMENTS

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

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]]