

A287769


{0>1, 1>110}transform of the infinite Fibonacci word A003849.


3



1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1
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OFFSET

1


COMMENTS

From Michel Dekking, Oct 11 2017: (Start)
(a(n)) is the homogeneous Sturmian sequence with slope r = (15 + sqrt(5))/22.
Note that (a(n)) can be obtained as the binary complement of the sequence a~ produced by applying the morphism gamma: 0>001, 1>0 to the binary complement x of the Fibonacci word A003849. One has x = A005614, the infinite Fibonacci word generated by 0>1, 1>10. Moreover, gamma can be written as a composition gamma = psi_3 psi_1 of the two elementary Sturmian morphisms psi_1: 0>01, 1>0 (standard Fibonacci), and psi_3: 0>0, 1>01. This implies immediately that a~ = gamma(x) is Sturmian, and to get the slope, use Lemma 2.2.18 in Lothaire, which gives that the Sturmian word b := psi_1(x) has slope (2phi)/(3phi) = (5sqrt(5))/10 (b = A221150), and next a~ = psi_3(b) has slope 1/(3+phi) = (7  sqrt(5))/22. Thus (a(n)) has slope 1 (7sqrt(5))/22 = (15 + sqrt(5))/22.
(End)
The algebraic conjugate t = (15  sqrt(5))/22 of r lies in (0,1), so by Allauzen's criterion, (a(n)) is NOT a fixed point of a morphism.  Michel Dekking, Oct 11 2017


LINKS

Clark Kimberling, Table of n, a(n) for n = 1..10000
M. Lothaire, Algebraic combinatorics on words, Cambridge University Press. Online publication date: April 2013; Print publication year: 2002.


FORMULA

a(n) = floor((n+1)*r)  floor(nr), where r = (15 + sqrt(5))/22.  Michel Dekking, Oct 11 2017


EXAMPLE

As a word, A003849 = 0100101001001010010100100..., and replacing each 0 by 1 and each 1 by 110 gives 11101111011101111011110111011110111011110...


MATHEMATICA

s = Nest[Flatten[# /. {0 > {0, 1}, 1 > {0}}] &, {0}, 10] (* A003849 *)
w = StringJoin[Map[ToString, s]]
w1 = StringReplace[w, {"0" > "1", "1" > "110"}]
st = ToCharacterCode[w1]  48 (* A287769 *)
Flatten[Position[st, 0]] (* A276855 *)
Flatten[Position[st, 1]] (* A287770 *)
Table[Floor[(n + 1) (15 + Sqrt[5]) / 22]  Floor[n (15 + Sqrt[5]) / 22], {n, 100}] (* Vincenzo Librandi, Ovt 15 2017 *)


PROG

(MAGMA) [Floor((n+1)*(15+Sqrt(5))/22)Floor(n*(15+Sqrt(5))/22): n in [1..100]]; // Vincenzo Librandi, Oct 15 2017


CROSSREFS

Cf. A005614, A221150, A276855, A287770.
Sequence in context: A210826 A307421 A299406 * A267866 A175087 A318924
Adjacent sequences: A287766 A287767 A287768 * A287770 A287771 A287772


KEYWORD

nonn,easy


AUTHOR

Clark Kimberling, Jun 03 2017


STATUS

approved



