A188068 a(n) = [n*r]-[k*r]-[n*r-k*r], where r=sqrt(3), k=1, [ ]=floor.

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

Offset

1

Comments

Sturmian word with slope alpha = sqrt(3)-1, and offset 0. Since alpha has a periodic continued fraction expansion with period 12, (a(n+1)) is the unique fixed point of the morphism 0 -> 110, 1 -> 1101. - Michel Dekking, Feb 06 2017

A275855(n) = R(a(n)) for n>1, where R is the mirror morphism R(0)=1, R(1)=0, This can be shown by induction on the iterates of the two morphisms generating the sequences. - Michel Dekking, Feb 07 2017

See also A188014.

References

J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, p. 286.

Links

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Formula

a(n) = A007538(n) - 2. - Reinhard Zumkeller, Feb 14 2012

Mathematica

r=3^(1/2); k=1;
t=Table[Floor[n*r]-Floor[(n-k)*r]-Floor[k*r], {n, 1, 220}]   (* A188068 *)
Flatten[Position[t, 0]]  (* A188069 *)
Flatten[Position[t, 1]]  (* A188070 *)

Prog

(Haskell)
a188068 = (subtract 2) . a007538  -- Reinhard Zumkeller, Feb 14 2012
(Python)
from gmpy2 import isqrt
def A188068(n):
    return int(isqrt(3*n**2) - isqrt(3*(n-1)**2)) - 1 # Chai Wah Wu, Oct 07 2016

Crossrefs

Cf. A188014, A188069, A188070.

Sequence in context: A155897 A368915 A144610 * A181632 A105565 A332814

Adjacent sequences: A188065 A188066 A188067 * A188069 A188070 A188071

Keyword

nonn

Author

Clark Kimberling, Mar 20 2011

Status

approved