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A144611 Sturmian word of slope sqrt(2). 1
0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

LINKS

Danny Rorabaugh, Table of n, a(n) for n = 0..10000

Mike Winkler, On the structure and the behaviour of Collatz 3n+ 1 sequences, 2014.

FORMULA

Conjecture: a(n) = floor((n+1)*log(3)/log(2)) - floor(n*log(3)/log(2)) - 1.

This is not true: Let b(n) = floor((n+1)*log(3)/log(2)) - floor(n*log(3)/log(2)) - 1. Then b(40) = 0, whereas a(40) = 1. This is the first term at which a(n) and b(n) disagree. - Danny Rorabaugh, Mar 14 2015

From Benoit Cloitre, Oct 16 2016: (Start)

Let u(n)=n+floor(sqrt(2)*n) (A003151) and v(n)=n+floor(n/sqrt(2)) (A003152) then u,v form a partition of the positive integers and we have for n>=1 a(u(n))=0 and a(v(n))=1.

Another way to construct the sequence: merge the sequences x(n)=2n^2+1 and y(n)=4n^2 (n>=1) into an increasing sequence z(n) which then begins: 3,4,9,16,19,33,36,51,64,73 (not in OEIS) . Then for n>=1 a(n)=z(n) modulo 2. (End)

MATHEMATICA

christoffel[s_, M_] := Module[{n, x = 1, y = 0, ans = {0}}, Do[If[y + 1 <= s*x, AppendTo[ans, 1]; y++, AppendTo[ans, 0]; x++], {n, 1, M}]; ans] (* or Sturmian word, Jean-Fran├žois Alcover, Sep 19 2016, A274170 *); christoffel[Sqrt[2], 105] (* Robert G. Wilson v, Feb 02 2017 *)

PROG

(Sage) #Generate the first n terms (plus a few) of the Sturmian word of slope a

def Sturmian(a, n):

..y = 0

..A = []

..while len(A)<=n:

....y += a

....A.extend([0]+[1]*(floor(y)-floor(y-a)))

..return A

Sturmian(sqrt(2), 104)

# Danny Rorabaugh, Mar 14 2015

(PARI) //to get N terms//

a(n)=if(n<1, 0, vecsort(concat(vector(floor(sqrt(2)*N), i, 2*i^2+1), vector(N, j, 4*j^2)))[n]%2) \\ Benoit Cloitre, Oct 16 2016

CROSSREFS

See A144595 for further details.

Sequence in context: A113217 A147781 A082446 * A130853 A115516 A090171

Adjacent sequences:  A144608 A144609 A144610 * A144612 A144613 A144614

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Jan 13 2009

STATUS

approved

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Last modified July 22 03:22 EDT 2017. Contains 289648 sequences.