A285830 Triangular array: row n shows the n+1 zero-one subwords of length n that occur in the infinite Fibonacci word A003849, in the order of occurrence.

0, 1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0

Offset

1

Comments

Exactly n+1 zero-one-words of length n occur as subwords of the infinite Fibonacci word w = A003849 = 01001010010010100101... For n = 0..5, they are listed here in the order of appearance.

n subwords of w

0 the empty word

1 0, 1

2 01, 10, 00

3 010 100 001 101

4 0100, 1001, 0010, 0101, 1010

5 01001, 10010, 00101, 01010, 10100, 00100

Links

Table of n, a(n) for n=1..86.

Example

Starting with n=1, take in order the zeros and ones in the triangle of words shown in Comments: 0, 1, 01, 10, 00, 010, 100, 001, 101, ... ; these are represented as 0,1,0,1,1,0,0,0,0,1,0,1,0,0,0,0,1,1,0,1,...

Mathematica

s = Nest[Flatten[# /. {0 -> {0, 1}, 1 -> {0}}] &, {0}, 10] (* A003849 *)
(w = Table[DeleteDuplicates[Partition[s, k, 1]], {k, Floor[Length[s/2]]}]) // ColumnForm (* A285830, array *)
Map[Sort, w] // ColumnForm (* A285831, array *)
w1 = Map[Sort, w] ;
Flatten[w]  (* A285830, sequence *)
Flatten[w1] (* A285831, sequence *)
(* Peter J. C. Moses, Apr 26 2017 *)

Crossrefs

Cf. A285831.

Sequence in context: A130853 A353810 A115516 * A288741 A341684 A327183

Adjacent sequences: A285827 A285828 A285829 * A285831 A285832 A285833

Keyword

nonn,easy,tabf

Author

Clark Kimberling, May 02 2017

Status

approved