login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A171588 Fixed point of the morphism 0->001, 1->0. Pell word. 24
0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

From Peter Bala, Nov 22 2013: (Start)

Sturmian word: equals the limit word S(infinity) where S(0) = 0, S(1) = 001 and for n >= 1, S(n+1) = S(n)S(n)S(n-1). See the examples below.

This sequence corresponds to the case k = 2 of the Sturmian word S_k(infinity) as defined in A080764. See A159684 for the case k = 1. (End)

Characteristic word with slope 1 - 1/sqrt(2). Since the characteristic word with slope 1-theta is the mirror image of the characteristic word with slope theta, a(n)= 1 - A080764(n) for all n. - Michel Dekking, Jan 31 2017

The positions of 0 comprise A001951 (Beatty sequence for sqrt(2)); the positions of 1 comprise A001952 (Beatty sequence for 2+sqrt(2)). - Clark Kimberling, May 11 2017

REFERENCES

J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, p. 284.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..5000

Scott Balchin and Dan Rust, Computations for Symbolic Substitutions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.4.1.

Jean Berstel and Juhani Karhumäki, Combinatorics on words-a tutorial. Bull. Eur. Assoc. Theor. Comput. Sci. EATCS, 79:178-228, 2003.

M. Lothaire, Combinatorics on Words.

Wikipedia, Sturmian word

FORMULA

a(n) = floor((n + 1)/(2 + sqrt(2))) - floor(n /(2 + sqrt(2))). - Peter Bala, Nov 22 2013

a(n) = floor((n+1)(1 - 1/sqrt(2)) - floor(n (1 - 1/sqrt(2)). - Michel Dekking, Jan 31 2017

EXAMPLE

From Peter Bala, Nov 22 2013: (Start)

The sequence of words S(n) begins

S(0) = 0

S(1) = 001

S(2) = 001 001 0

S(3) = 0010010 0010010 001

S(4) = 00100100010010001 00100100010010001 0010010.

The lengths of the words are [1, 3, 7, 17, 41, ...] = A001333 (apart from the initial term).  (End)

MAPLE

Digits := 50: u := evalf(2 + sqrt(2)): A171588 := n->floor((n+1)/u) - floor(n/u): seq(A171588(n), n = 1..80); # Peter Bala, Nov 22 2013

MATHEMATICA

Table[Floor[(n + 1) (1 - 1/Sqrt[2]) - Floor[n (1 - 1/Sqrt[2])]], {n, 100}] (* Vincenzo Librandi, Jan 31 2017 *)

Nest[Flatten[# /. {0 -> {0, 0, 1}, 1 -> {0}}] &, {0}, 6] (* Clark Kimberling, May 11 2017 *)

PROG

(MAGMA) [Floor((n+1)*(1-1/Sqrt(2))-Floor(n*(1-1/Sqrt(2)))): n in [1..100]]; // Vincenzo Librandi, Jan 31 2017

CROSSREFS

Cf. A000129, A001333, A001951, A001952, A003849, A080764, A159684.

Sequence in context: A049320 A080887 A099395 * A131531 A022003 A144604

Adjacent sequences:  A171585 A171586 A171587 * A171589 A171590 A171591

KEYWORD

nonn,easy

AUTHOR

Alexis Monnerot-Dumaine (alexis.monnerotdumaine(AT)gmail.com), Dec 12 2009

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified October 17 10:50 EDT 2017. Contains 293469 sequences.