A178992 Ordered list in decimal notation of the subwords (with leading zeros omitted) appearing in the infinite Fibonacci word A005614 (0->1 & 1->10).

0, 1, 2, 3, 5, 6, 10, 11, 13, 21, 22, 26, 27, 43, 45, 53, 54, 86, 90, 91, 107, 109, 173, 181, 182, 214, 218, 346, 347, 363, 365, 429, 437, 693, 694, 726, 730, 858, 859, 875, 1387, 1389, 1453, 1461, 1717, 1718, 1750, 2774, 2778, 2906, 2907, 2923, 3435, 3437, 3501

Offset

1,3

Comments

The definition mentions the Fibonacci word A005614. Note that the official Fibonacci word is A003849, which would give a different list, namely, the 2's-complement of the present list. - N. J. A. Sloane, Jan 12 2011

References

J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003.

Links

T. D. Noe, Table of n, a(n) for n = 1..1015

T. D. Noe, Table of n, a(n) for n = 1..1652 [The first 1652 terms written in binary, including leading zeros. Not a b-file.]

Example

The Fibonacci word has a minimal complexity, i.e., for any n there are n+1 distinct subwords of length n (see for example Allouche and Shallit).
E.g. for n=1 they are '0' and '1', for n=2 '01', '10' and '11' or, in decimal notation '1','2',and '3'.
Some subwords prefixed with '0' have the same decimal value as shorter ones, but there is no real ambiguity as double zeros do not appear in the infinite Fibonacci word.

Mathematica

iter=8; f=Nest[Flatten[# /. {0 -> {1}, 1 -> {1, 0}}] &, {1}, iter]; u={}; n=1; While[lst={}; k=0; While[num=FromDigits[Take[f, {1, n}+k], 2]; lst=Union[lst, {num}]; Length[lst]<n+1 && k<Length[f]-n, k++]; Length[lst]==n+1, u=Union[u, lst]; n++]; u

Crossrefs

Cf. A003849, A005614, A171676, A179969.

Sequence in context: A297571 A316503 A316494 * A076474 A324968 A255059

Adjacent sequences: A178989 A178990 A178991 * A178993 A178994 A178995

Keyword

nonn,nice,base

Author

Alexandre Losev, Jan 03 2011

Extensions

Definition clarified by N. J. A. Sloane, Jan 10 2011

Status

approved