A285681 Positions of 0 in A285680.

1, 3, 4, 6, 8, 9, 11, 13, 15, 16, 18, 20, 21, 23, 25, 27, 28, 30, 32, 34, 35, 37, 39, 40, 42, 44, 46, 47, 49, 51, 52, 54, 56, 58, 59, 61, 63, 65, 66, 68, 70, 71, 73, 75, 77, 78, 80, 82, 84, 85, 87, 89, 90, 92, 94, 96, 97, 99, 101, 102, 104, 106, 108, 109

Offset: 1

Author

Clark Kimberling, May 11 2017

Keywords

• nonn
• easy

Comments

A 3-way partition of the positive integers, by positions of 0, 1, 2 in A285680:

A285681: positions of 0; slope t = (15+sqrt(5))/10;

A285682: positions of 1; slope u = 4+sqrt(5);

A285683: positions of 2; slope v = (1+3*sqrt(5))/2;

where 1/t + 1/u + 1/v = 1. Conjecture: a(n) - a(n-1) is in {1,2} for n >= 2.

Links

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

Crossrefs

Cf. A003849, A285680, A285682, A285683.

Programs

Mathematica

s = Nest[Flatten[# /. {0 -> {0, 1}, 1 -> {0}}] &, {0}, 13] ; (* A003849 *)
w = StringJoin[Map[ToString, s]]
w1 = StringReplace[w, {"1010" -> "2"}]
st = ToCharacterCode[w1] - 48; (* A285680 *)
Flatten[Position[st, 0]];  (* A285681 *)
Flatten[Position[st, 1]];  (* A285682 *)
Flatten[Position[st, 2]];  (* A285683 *)