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A230603
Generalized Fibonacci word. Binary complement of A221150.
2
1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1
OFFSET
0
COMMENTS
Define strings S(0) = 1, S(1)= 110, thereafter S(n) = S(n-1)S(n-2); this sequence is the limit string S(infinity). See the examples below.
FORMULA
a(n) = floor((n + 2)/(3 - phi)) - floor((n + 1)/(3 - phi)), where phi = 1/2*(1 + sqrt(5)) is the golden ratio.
If we read the sequence as the decimal constant C = 0.11011 10110 11101 11011 01110 ... then C = sum {n >= 1} 1/10^floor(n*(3 - phi)).
9*C has the simple continued fraction expansion [0; 1, 110, 10^1, 10^3, 10^4, 10^7, ..., 10^Lucas(n), ...].
EXAMPLE
S(0) = 1
S(1) = 110
S(2) = 110 1
S(3) = 1101 110
S(4) = 1101110 1101
S(5) = 11011101101 1101110
The sequence of word lengths [1, 2, 4, 7, 11, 18, ...] is A000204.
MAPLE
Digits := 50: u := evalf((5-sqrt(5))/2): A230603 := n->floor((n+2)/u)-floor((n+1)/u): seq(A230603(n), n = 0..80);
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Peter Bala, Nov 22 2013
STATUS
approved