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A188221

a(n) = [n*r + k*r]-[n*r]-[k*r], 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

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OFFSET

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

Alternatively, express n-1 in the greedy numeration system based on 1,4,17,72,... (that is, A001076), and run it through the automaton depicted in the links. If it ends in a state labeled i/j, the n-th term of the sequence is j. - Jeffrey Shallit, Mar 03 2025

LINKS

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

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

Jeffrey Shallit, Automaton for the sequence

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