A285954 Positions of 1 in A285952; complement of A285953.

1, 2, 4, 6, 7, 9, 10, 11, 13, 15, 16, 17, 19, 20, 22, 24, 25, 27, 28, 29, 31, 32, 34, 36, 37, 38, 40, 42, 43, 45, 46, 47, 49, 51, 52, 53, 55, 56, 58, 60, 61, 62, 64, 66, 67, 69, 70, 71, 73, 74, 76, 78, 79, 81, 82, 83, 85, 87, 88, 89, 91, 92, 94, 96, 97, 99

Offset: 1

Author

Clark Kimberling, May 05 2017

Keywords

• nonn
• easy

Comments

Conjecture: 3n/2 - a(n) is in {0, 1/2, 1} for n >= 1.

Proof of the conjecture: Let t=A010060 be the Thue-Morse sequence. Every pair t(2n-1),t(2n) is either 01 or 10. Since 01 and 10 map to 110 and 101 under the transform, which both have length 3, it follows that a(2n+1) = 3n+1, and a(2n) = 3n if t(2n)=0, a(2n) = 3n-1 if t(2n)=1 for n=1,2,..., and so certainly 3n/2 - a(n) is 0, 1/2 or 1. - Michel Dekking, Jan 05 2018

Examples

			As a word, A285952 = 110101101110101..., in which 1 is in positions 1,2,4,6,7,9,...
		

Links

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

Crossrefs

Cf. A010060, A005614, A285950, A285952, A285953.

Programs

Mathematica

s = Nest[Flatten[# /. {0 -> {0, 1}, 1 -> {1, 0}}] &, {0}, 7]  (* Thue-Morse, A010060 *)
w = StringJoin[Map[ToString, s]]
w1 = StringReplace[w, {"0" -> "1", "1" -> "10"}]  (* A285952, word *)
st = ToCharacterCode[w1] - 48 (* A285952, sequence *)
Flatten[Position[st, 0]]  (* A285953 *)
Flatten[Position[st, 1]]  (* A285954 *)

Python

def A285954(n): return n+(n>>1)-(0 if n&1 else (n-1).bit_count()&1) # Chai Wah Wu, Nov 17 2024

Formula

a(2n+1) = 3n+1, a(2n) = 3n - A010060(2n) - Michel Dekking, Jan 05 2018

a(n) = n+floor(n/2) if n is odd and a(n) = n+floor(n/2)-A010060(n-1) otherwise. - Chai Wah Wu, Nov 17 2024