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
Examples
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.
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.]
Programs
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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]
Extensions
Definition clarified by N. J. A. Sloane, Jan 10 2011
Comments