A285679 Positions of 2 in A285677.

3, 5, 10, 12, 17, 22, 24, 29, 31, 36, 41, 43, 48, 53, 55, 60, 62, 67, 72, 74, 79, 81, 86, 91, 93, 98, 103, 105, 110, 112, 117, 122, 124, 129, 134, 136, 141, 143, 148, 153, 155, 160, 162, 167, 172, 174, 179, 184, 186, 191, 193, 198, 203, 205, 210, 212, 217

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 A285677:

A285678: positions of 0; slope t = (4+sqrt(5))/2;

A182761: positions of 1; slope u = (7-sqrt(5))/2;

A285679: 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 {2,5} for n>=2.

See A285683 for a proof of this conjecture. - Michel Dekking, Oct 09 2018

a(n) = A285683(n-1) for n>1, see A285683 for a proof. - Michel Dekking, Oct 09 2018

Links

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

Crossrefs

Cf. A003849, A284620, A285678, A182761, A285679.

Programs

Mathematica

s = Nest[Flatten[# /. {0 -> {0, 1}, 1 -> {0}}] &, {0}, 13] ; (* A003849 *)
w = StringJoin[Map[ToString, s]]
w1 = StringReplace[w, {"0010" -> "2"}]
st = ToCharacterCode[w1] - 48; (* A285677 *)
Flatten[Position[st, 0]];  (* A285678 *)
Flatten[Position[st, 1]];  (* A182761 *)
Flatten[Position[st, 2]];  (* A285679 *)

Formula

a(n) = 3*floor((n-1)*phi) - n + 4