A188221 [nr+kr]-[nr]-[kr], where r=sqrt(5), k=1, [ ]=floor.

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

Offset

1

Comments

Differs from A188187 in number of initial zeros. This sequence yields right-shift sums (introduced at A187950), and A188187 yields left-shift sums (A188014).

(a(n)) is the Sturmian sequence with slope sqrt(5)-2. This number has continued fraction expansion [0; 4,4,4, ...]. It is therefore fixed point of a morphism sigma, which can be found with the Crisp et al link: sigma is given by 0 -> 0001, 1 -> 00010. - Michel Dekking, Aug 02 2017

Links

Table of n, a(n) for n=1..148.

D. Crisp, W. Moran, A. Pollington, P. Shiue, Substitution invariant cutting sequences, Journal de théorie des nombres de Bordeaux 5, (1993), p. 123-137.

Formula

a(n) = [nr+r]-[nr]-[r], where r=sqrt(5).

Mathematica

r=5^(1/2); k=1;
seqA=Table[Floor[n*r+k*r]-Floor[n*r]-Floor[k*r], {n, 1, 220}]   (* A188221 *)
Flatten[Position[seqA, 0] ]   (* A188222 *)
Flatten[Position[seqA, 1] ]   (* A004976 *)

Prog

(Python)
from sympy import integer_nthroot
def A188221(n): return integer_nthroot(5*(n+1)**2, 2)[0]-integer_nthroot(5*n**2, 2)[0]-2 # Chai Wah Wu, Mar 16 2021

Crossrefs

Cf. A187950, A188222, A004976, A188014, A188187, A188188, A004958.

Sequence in context: A105563 A188291 A287372 * A011765 A342023 A342024

Adjacent sequences: A188218 A188219 A188220 * A188222 A188223 A188224

Keyword

nonn

Author

Clark Kimberling, Mar 24 2011

Status

approved