A274170 -id:A274170 - cp's OEIS frontend

A144611 Sturmian word of slope 2-sqrt(2).

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

Offset: 0

N. J. A. Sloane, Jan 13 2009

nonn

Old name was: Sturmian word of slope 2.
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 the OEIS). Then for n >= 1, a(n) = z(n) mod 2. (End)
From Michel Dekking, Feb 16 2020: (Start)
This sequence is a Sturmian sequence s(alpha,rho) with slope alpha = 2-sqrt(2), and intercept rho = 0.
In general, one passes from slope alpha to slope 1-alpha by exchanging 0 and 1. It therefore follows from the Comments of A006337 that (a(n+1)) is the unique fixed point of the morphism 0 -> 101, 1 -> 10. (End)

Danny Rorabaugh, Table of n, a(n) for n = 0..10000
M. Lothaire, Algebraic combinatorics on words, Cambridge University Press. Online publication date: April 2013; Print publication year: 2002.
Mike Winkler, On the structure and the behaviour of Collatz 3n+ 1 sequences, arXiv:1412.0519 [math.GM], 2014.

See A144595 for further details. Cf. A006337, A074840.

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-François Alcover, Sep 19 2016, A274170 *); christoffel[Sqrt[2], 105] (* Robert G. Wilson v, Feb 02 2017 *)

\\ 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

#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

a(n) = floor((n+1)*alpha) - floor(n*alpha), where alpha = 2-sqrt(2). - Michel Dekking, Feb 16 2020

Name corrected by Michel Dekking, Feb 16 2020

A144609 Sturmian word of slope Pi.

Original entry on oeis.org

0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1

Offset: 0

N. J. A. Sloane, Jan 13 2009

nonn

A063438 seems to contain the run lengths of 1's. - R. J. Mathar, May 30 2025

See A144595 for further details.
Seems to be very similar to A070127. Is this a coincidence?
Cf. A063438, A076539 (partial sums).

Digits := 500 :
x :=1 ;
y :=0 ;
slop := Pi ;
printf("0,") ;
for n from 1 to 300 do
    if evalf((y+1)/x-slop) > 0 then
        x := x+1 ;
        printf("0,") ;
    else
        y := y+1 ;
        printf("1,") ;
    end if;
end do: # R. J. Mathar, May 30 2025

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]; christoffel[Pi, 105] (* Robert G. Wilson v, Feb 02 2017, after Jean-François Alcover, Sep 19 2016, A274170 *)

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A144611 Sturmian word of slope 2-sqrt(2).

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