A288998 Positions of 0 in A288997; complement of A288999.

1, 2, 4, 5, 6, 8, 9, 11, 12, 13, 15, 16, 18, 19, 21, 22, 23, 25, 26, 28, 29, 30, 32, 33, 35, 36, 38, 39, 40, 42, 43, 45, 46, 47, 49, 50, 52, 53, 54, 56, 57, 59, 60, 62, 63, 64, 66, 67, 69, 70, 71, 73, 74, 76, 77, 79, 80, 81, 83, 84, 86, 87, 88, 90, 91, 93

Offset: 1

Author

Clark Kimberling, Jun 25 2017

Keywords

• nonn
• easy

Comments

Conjecture: a(n)/n -> sqrt(2), and 0 < sqrt(2) - a(n)/n < -1 + sqrt(2) for n >= 1.

Links

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

Crossrefs

Cf. A288997, A288999.

Programs

Mathematica

z = 10; (* number of iterates *)
s = {0, 0}; w[0] = StringJoin[Map[ToString, s]];
w[n_] := StringReplace[w[n - 1], {"00" -> "0010", "01" -> "001", "10" -> "001"}]
TableForm[Table[w[n], {n, 0, 10}]]
st = ToCharacterCode[w[z]] - 48   (* A288997 *)
Flatten[Position[st, 0]]  (* A288998 *)
Flatten[Position[st, 1]]  (* A288999 *)

Formula

a(n) = round((n-1/2)*sqrt(2)) (conjectured). - Chai Wah Wu, Feb 06 2025