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A189661 Fixed point of the morphism 0->010, 1->10 starting with 0. 11
0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET
1
COMMENTS
From Michel Dekking, Oct 20 2018: (Start)
Let alpha=(3-sqrt(5))/2 be the 'small golden mean'.
The following two facts follow from Proposition 2 and Theorem 2 in my paper on substitution invariant words.
(I) The sequence {a(n)} is the inhomogeneous Sturmian sequence
s'(alpha,1-alpha) = (ceiling(n*alpha+1-alpha)-ceiling((n-1)*alpha+1-alpha)).
(II) The other fixed point of 0->010, 1->10 is the inhomogeneous Sturmian sequence
A289034 = s(alpha,1-alpha) = (floor(n*alpha+1-alpha)-floor((n-1)*alpha+1-alpha)).
a(n) = A289034(n) for all n > 2, but a(1),a(2) = 0,1; A289034(1),A289034(2) = 1,0.
(End)
REFERENCES
Bernardino, André, Rui Pacheco, and Manuel Silva. "Coloring factors of substitutive infinite words." Discrete Mathematics 340.3 (2017): 443-451. See Example 2.
LINKS
A. Bernardino, M. Silva, R. Pacheco, Coloring factors of substitutive infinite words, arXiv:1605.09343 [math.CO], 2016.
Michel Dekking, Substitution invariant Sturmian words and binary trees, Integers, Electronic Journal of Combinatorial Number Theory 18A (2018), #A17.
EXAMPLE
0->010->01010010->1001001010010->...
MATHEMATICA
t = Nest[Flatten[# /. {0->{0, 1, 0}, 1->{1, 0}}] &, {0}, 5] (*A189661*)
f[n_] := t[[n]]
Flatten[Position[t, 0]] (*A189662*)
Flatten[Position[t, 1]] (*A026356*)
s[n_] := Sum[f[i], {i, 1, n}]; s[0] = 0;
Table[s[n], {n, 1, 120}] (*A189663*)
SubstitutionSystem[{0->{0, 1, 0}, 1->{1, 0}}, {0}, {5}][[1]] (* Harvey P. Dale, Mar 29 2022 *)
CROSSREFS
Sequence in context: A285966 A215530 A241422 * A145573 A092202 A358846
KEYWORD
nonn
AUTHOR
Clark Kimberling, Apr 25 2011
EXTENSIONS
Name corrected by Michel Dekking, Oct 18 2018
STATUS
approved

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Last modified February 21 07:47 EST 2024. Contains 370219 sequences. (Running on oeis4.)