A285520 - OEIS

A285520

Positions of 1 in A285518; complement of A285519.

1, 2, 4, 6, 7, 8, 10, 12, 13, 14, 16, 18, 19, 20, 22, 24, 26, 28, 30, 32, 33, 34, 36, 38, 39, 40, 42, 44, 46, 48, 50, 52, 53, 54, 56, 58, 59, 60, 62, 64, 65, 66, 68, 70, 71, 72, 74, 76, 77, 78, 80, 82, 83, 84, 86, 88, 90, 92, 94, 96, 97, 98, 100, 102, 103

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OFFSET

1,2

COMMENTS

Conjecture: a(n)/n -> (1+sqrt(5))/2 = golden ratio (A001622).

Proof of this conjecture: the limit equals 1/f1, where f1 is the frequency of 1 in (a(n)), and 1/f1 = phi = golden ratio. See A285519 for more details. - Michel Dekking, Feb 22 2021

LINKS

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

EXAMPLE

As a word, A285518 = 11010111010111010111..., in which 1 is in positions 1,2,4,6,7,...

MATHEMATICA

s = Nest[Flatten[# /. {0 -> {1, 1}, 1 -> {0, 0, 1, 1}}] &, {0}, 7] (* A285504 *)

w = StringJoin[Map[ToString, s]]

w1 = StringReplace[w, {"11" -> "1", "00" -> "0"}]

s1 = ToCharacterCode[w1] - 48 (* A285518 *)

Flatten[Position[s1, 0]] (* A285519 *)

Flatten[Position[s1, 1]] (* A285520 *)

CROSSREFS