A285678 -id:A285678 - cp's OEIS frontend

A285677 {0010->2}-transform of the infinite Fibonacci word A003849.

0, 1, 2, 1, 2, 0, 1, 0, 1, 2, 1, 2, 0, 1, 0, 1, 2, 0, 1, 0, 1, 2, 1, 2, 0, 1, 0, 1, 2, 1, 2, 0, 1, 0, 1, 2, 0, 1, 0, 1, 2, 1, 2, 0, 1, 0, 1, 2, 0, 1, 0, 1, 2, 1, 2, 0, 1, 0, 1, 2, 1, 2, 0, 1, 0, 1, 2, 0, 1, 0, 1, 2, 1, 2, 0, 1, 0, 1, 2, 1, 2, 0, 1, 0, 1, 2

Offset: 1

Clark Kimberling, May 11 2017

As a word, A003849 = 01001010010010100..., and replacing each 0010 by 2 gives 0121201012120101201012120101212010...

Warning: "replacing each 0010 by 2" means "replacing each 0010 by 2 from left to right, consecutively". The result is that the word a(8)...a(14)=0010010 in A003849 is replaced by 201, not by 22. - Michel Dekking, Aug 27 2018

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

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

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 - conjectured *)
Flatten[Position[st, 2]];  (* A285679 *)

A285679 Positions of 2 in A285677.

Original entry on oeis.org

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

Clark Kimberling, May 11 2017

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

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

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

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 *)

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

cp's OEIS Frontend

A285677 {0010->2}-transform of the infinite Fibonacci word A003849.

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