A230603 Generalized Fibonacci word. Binary complement of A221150.

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.

Links

Table of n, a(n) for n=0..80.

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

Cf. A000204, A003849, A005614, A221150.

Sequence in context: A188090 A004547 A285358 * A229343 A085369 A188082

Adjacent sequences: A230600 A230601 A230602 * A230604 A230605 A230606

Keyword

nonn,easy

Author

Peter Bala, Nov 22 2013

Status

approved