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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 (list; graph; refs; listen; history; text; internal format)
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 (2-phi)/(3-phi) = (5-sqrt(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- (7-sqrt(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

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Last modified October 21 03:24 EDT 2019. Contains 328291 sequences. (Running on oeis4.)