A285831 Triangular array: row n shows the n+1 zero-one subwords of length n that occur in the infinite Fibonacci word A003849, in lexicographic order.

0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 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 lexicographic order. (See A285830 for a listing in order of appearance in w.)

n subwords of w

0 the empty word

1 0, 1

2 00, 01, 10

3 001, 010, 100, 101

4 0010, 0100, 0101, 1001, 1010

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

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, 00, 01, 10, 001, 010, 100, 101, ...; represent these as 0,1,0,0,0,1,1,0,0,0,1,0,1,0,1,0,0,1,0,1,...

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. A285830.

Sequence in context: A286052 A342910 A341258 * A188294 A079101 A334941

Adjacent sequences: A285828 A285829 A285830 * A285832 A285833 A285834

Keyword

nonn,easy,tabf

Author

Clark Kimberling, May 02 2017

Status

approved