A287769 {0->1, 1->110}-transform of the infinite Fibonacci word A003849.

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

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