A285953 Positions of 0 in A285952; complement of A285954.

3, 5, 8, 12, 14, 18, 21, 23, 26, 30, 33, 35, 39, 41, 44, 48, 50, 54, 57, 59, 63, 65, 68, 72, 75, 77, 80, 84, 86, 90, 93, 95, 98, 102, 105, 107, 111, 113, 116, 120, 123, 125, 128, 132, 134, 138, 141, 143, 147, 149, 152, 156, 158, 162, 165, 167, 170, 174, 177

Offset

1,1

Comments

Conjecture: 3n - a(n) is in {0, 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(n) = 3n-1+t(2n) for n=1,2,..., and so certainly 3n - a(n) is 0 or 1. - Michel Dekking, Jan 05 2018

Links

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

Formula

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

Example

As a word, A285952 = 110101101110101..., in which 0 is in positions  3,5,8,12,...

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 *)

Prog

(Python)
def A285953(n): return 3*n-(((n<<1)-1).bit_count()&1^1) # Chai Wah Wu, Nov 18 2024

Crossrefs

Cf. A010060, A005614, A285950, A285952, A285954.

Sequence in context: A342779 A100464 A114891 * A287380 A339389 A348236

Adjacent sequences: A285950 A285951 A285952 * A285954 A285955 A285956

Keyword

nonn,easy

Author

Clark Kimberling, May 05 2017

Status

approved