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)).
(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
KEYWORD
nonn
AUTHOR
Clark Kimberling, Apr 25 2011
EXTENSIONS
Name corrected by Michel Dekking, Oct 18 2018
STATUS
approved