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%I #66 Apr 13 2020 01:50:02
%S 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,
%T 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,
%U 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
%N Sturmian word of slope 2-sqrt(2).
%C Old name was: Sturmian word of slope 2.
%C Conjecture: a(n) = floor((n+1)*log(3)/log(2)) - floor(n*log(3)/log(2)) - 1.
%C 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
%C From _Benoit Cloitre_, Oct 16 2016: (Start)
%C 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.
%C 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)
%C From _Michel Dekking_, Feb 16 2020: (Start)
%C This sequence is a Sturmian sequence s(alpha,rho) with slope alpha = 2-sqrt(2), and intercept rho = 0.
%C 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)
%H Danny Rorabaugh, <a href="/A144611/b144611.txt">Table of n, a(n) for n = 0..10000</a>
%H M. Lothaire, <a href="http://tomlr.free.fr/Math%E9matiques/Fichiers%20Claude/Auteurs/aaaDivers/Lothaire%20-%20Algebraic%20Combinatorics%20On%20Words.pdf">Algebraic combinatorics on words</a>, Cambridge University Press. Online publication date: April 2013; Print publication year: 2002.
%H Mike Winkler, <a href="https://arxiv.org/abs/1412.0519">On the structure and the behaviour of Collatz 3n+ 1 sequences</a>, arXiv:1412.0519 [math.GM], 2014.
%F a(n) = floor((n+1)*alpha) - floor(n*alpha), where alpha = 2-sqrt(2). - _Michel Dekking_, Feb 16 2020
%t 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 *)
%o (Sage) #Generate the first n terms (plus a few) of the Sturmian word of slope a
%o def Sturmian(a,n):
%o y = 0
%o A = []
%o while len(A)<=n:
%o y += a
%o A.extend([0]+[1]*(floor(y)-floor(y-a)))
%o return A
%o Sturmian(sqrt(2),104)
%o # _Danny Rorabaugh_, Mar 14 2015
%o (PARI) \\ to get N terms
%o 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
%Y See A144595 for further details. Cf. A006337.
%K nonn
%O 0,1
%A _N. J. A. Sloane_, Jan 13 2009
%E Name corrected by _Michel Dekking_, Feb 16 2020
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