A189820 a(3*k-2) = a(k), a(3*k-1) = a(k), a(3*k) = 1 for k >= 1, starting with a(1) = 0.

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

Offset

1

Comments

Together with A189816, this sequence completes the list of "type 3,3" sequences described at A189628.

Alternate definition: a(n) = 1 if the ternary representation of n-1 has at least one "2" in it. Otherwise, a(n) = 0. Partial sums are given by A081610. - Nathaniel Johnston, May 17 2011

From Gabriele Fici, Mar 10 2024: (Start)

Fixed point of the substitution 0->001, 1->111.

An example of a Rote sequence, that is, it has exactly 2m distinct contiguous subsequences of length m, for every m >= 1. For example, there are 6 contiguous subsequences of length 3, namely 001, 010, 011, 100, 110, 111. (End)

Links

Antti Karttunen, Table of n, a(n) for n = 1..13122

Mazen Khodier, New Methods for Analyzing the Properties of Automatic Sequences, Master's Thesis, Univ. Waterloo (Canada 2026). See p. 27, Table 5.1.

Günter Rote, Sequences With Subword Complexity 2n, Journal of Number Theory, Volume 46, Issue 2, February 1994, pp. 196-213.

Index entries for characteristic functions

Mathematica

a[1] = 0; h = 180;
Table[a[3 k - 2] = a[k], {k, 1, h}];
Table[a[3 k - 1] = a[k], {k, 1, h}];
Table[a[3 k] = 1, {k, 1, h}];
Table[a[n], {n, 1, h}]   (* this sequence *)
Flatten[Position[%, 0]]  (* A003278 *)
Flatten[Position[%%, 1]] (* A189822 *)

Prog

(PARI) a(n)=if(n<6, n==3, if(n%3, a(n\3+1), 1)) \\ Charles R Greathouse IV, Aug 11 2011
(Scheme) (define (A189820 n) (cond ((= 1 n) 0) ((zero? (modulo n 3)) 1) (else (A189820 (+ 1 (/ (- n (modulo n 3)) 3)))))) ;; Antti Karttunen, Aug 18 2017
(Python)
from gmpy2 import digits
def A189820(n):
    l = (s:=digits(n-2, 3)).find('2')
    if l >= 0: s = s[:l]+'1'*(len(s)-l)
    m = (t:=digits(n-1, 3)).find('2')
    if m >= 0: t = t[:m]+'1'*(len(t)-m)
    return 1+int(s, 2)-int(t, 2) # Chai Wah Wu, Dec 05 2024

Crossrefs

Cf. A081610, A189628, A003278, A189822.

Sequence in context: A284388 A289174 A059125 * A111406 A353463 A383418

Adjacent sequences: A189817 A189818 A189819 * A189821 A189822 A189823

Keyword

nonn,easy

Author

Clark Kimberling, Apr 28 2011

Extensions

Typo in definition fixed by Reinhard Zumkeller, Aug 11 2011

Status

approved