A144595 Christoffel word of slope 4/7.

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

Offset: 0

Author

N. J. A. Sloane, Jan 13 2009

Keywords

• nonn
• easy

Comments

The Christoffel word (or path) of slope s is defined as follows.

Start at (0,0) in the 2-dimensional integer lattice and move up if possible, otherwise right, always keeping below or on the line y = s*x. Write down 0 with a horizontal move, 1 for a vertical move. The first move is necessarily horizontal, so the sequence always begins with 0.

If s is irrational this is called a Sturmian word.

If the first 9 terms are deleted we get the "Upper Christoffel word of slope 4/7" (see Berstal et al., p. 6, Fig. 2).

The length of a Christoffel word of fraction a/b > 0 is a + b, with a ones. - David A. Corneth, Sep 19 2016

References

• J. Berstel et al., Combinatorics on Words: Christoffel Words and Repetitions in Words, Amer. Math. Soc., 2008.

Links

• J. Berstel et al., Combinatorics on Words: Christoffel Words and Repetitions in Words

Crossrefs

Cf. A144596-A144608.

Programs

Maple

christoffel:=proc(s,M) local n,x,y,ans;
ans:=[0]; x:=1; y:=0;
for n from 1 to M do
if y+1 <= s*x then ans:=[op(ans),1]; y:=y+1; else ans:=[op(ans),0]; x:=x+1; fi;
od: ans; end; christoffel(4/7,120);

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]; christoffel[4/7, 120] (* Jean-François Alcover, Sep 19 2016, adapted from Maple *)

PARI

\\ Christoffel word for nonnegative rational f.
Cword(f) = {my(n = numerator(f), d = denominator(f), v = vector(n + d), c, s, t = 1, i = 1); v[#v] = 1; while(t<=#v-4, i++; c=(i*f>=s+1); if(c, i-=2; s++, t++); v[t+2]=c); v}
a(n) = my(c = Cword(4/7)); c[(n-1)%#c+1] \\ David A. Corneth, Sep 19 2016

Formula

Period 11: 0,0,1,0,0,1,0,0,1,0,1.

a(n) = a(n-11).

G.f. -x^2*(1+x^3+x^6+x^8) / ( (x-1)*(1+x^10+x^9+x^8+x^7+x^6+x^5+x^4+x^3+x^2+x) ). - R. J. Mathar, Jul 09 2013