A285830 - OEIS

%I #9 May 05 2018 04:17:54

%S 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,

%T 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,

%U 0,0,0,1,0,0,1,0,1,0,0,1,0,1,0,0,1,0

%N 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.

%C 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.

%C n     subwords of w

%C 0     the empty word

%C 1     0, 1

%C 2     01, 10, 00

%C 3     010 100 001 101

%C 4     0100, 1001, 0010, 0101, 1010

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

%e 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,...

%t s = Nest[Flatten[# /. {0 -> {0, 1}, 1 -> {0}}] &, {0}, 10] (* A003849 *)

%t (w = Table[DeleteDuplicates[Partition[s, k, 1]], {k, Floor[Length[s/2]]}]) // ColumnForm (* A285830, array *)

%t Map[Sort, w] // ColumnForm (* A285831, array *)

%t w1 = Map[Sort, w] ;

%t Flatten[w]  (* A285830, sequence *)

%t Flatten[w1] (* A285831, sequence *)

%t (* _Peter J. C. Moses_, Apr 26 2017 *)

%Y Cf. A285831.

%K nonn,easy,tabf

%O 1

%A _Clark Kimberling_, May 02 2017