A053838 a(n) = (sum of digits of n written in base 3) modulo 3.

0, 1, 2, 1, 2, 0, 2, 0, 1, 1, 2, 0, 2, 0, 1, 0, 1, 2, 2, 0, 1, 0, 1, 2, 1, 2, 0, 1, 2, 0, 2, 0, 1, 0, 1, 2, 2, 0, 1, 0, 1, 2, 1, 2, 0, 0, 1, 2, 1, 2, 0, 2, 0, 1, 2, 0, 1, 0, 1, 2, 1, 2, 0, 0, 1, 2, 1, 2, 0, 2, 0, 1, 1, 2, 0, 2, 0, 1, 0, 1, 2, 1, 2, 0, 2, 0, 1, 0, 1, 2, 2, 0, 1, 0, 1, 2, 1, 2, 0, 0, 1, 2, 1, 2, 0

Offset: 0

Author

Henry Bottomley, Mar 28 2000

Keywords

• base
• nonn

Comments

Start with 0, repeatedly apply the morphism 0->012, 1->120, 2->201. This is a ternary version of the Thue-Morse sequence A010060. See Brlek (1989). - N. J. A. Sloane, Jul 10 2012

A090193 is generated by the same mapping starting with 1. A090239 is generated by the same mapping starting with 2. - Andrey Zabolotskiy, May 04 2016

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..2000
• S. Brlek, Enumeration of factors in the Thue-Morse word, Discrete Applied Math. 24 (1989), 83-96.
• Arthur Dolgopolov, Equitable Sequencing and Allocation Under Uncertainty, Preprint, 2016.
• Glen Joyce C. Dulatre, Jamilah V. Alarcon, Vhenedict M. Florida, and Daisy Ann A. Disu, On Fractal Sequences, DMMMSU-CAS Science Monitor (2016-2017) Vol. 15 No. 2, 109-113.
• Michael Gilleland, Some Self-Similar Integer Sequences
• Michel Rigo, Relations on words, arXiv preprint arXiv:1602.03364 [cs.FL], 2016. See Example 17.
• Robert Walker, Self Similar Sloth Canon Number Sequences
• Index entries for sequences that are fixed points of mappings

Crossrefs

Cf. A004128, A010060, A053837, A053839-A053844.

Equals A026600(n+1) - 1.

Programs

Maple

A053838 := proc(n)
    add(d,d=convert(n,base,3)) ;
    modp(%,3) ;
end proc:
seq(A053838(n),n=0..100) ; # R. J. Mathar, Nov 04 2017

Mathematica

Nest[ Flatten[ # /. {0 -> {0, 1, 2}, 1 -> {1, 2, 0}, 2 -> {2, 0, 1}}] &, {0}, 7] (* Robert G. Wilson v, Mar 08 2005 *)

PARI

a(n) = vecsum(digits(n, 3)) % 3; \\ Michel Marcus, May 04 2016

Python

from sympy.ntheory import digits
def A053838(n): return sum(digits(n,3)[1:])%3 # Chai Wah Wu, Feb 28 2025

Formula

a(n) = A010872(A053735(n)) =(n+a(floor[n/3])) mod 3. So one can construct sequence by starting with 0 and mapping 0->012, 1->120 and 2->201 (e.g. 0, 012, 012120201, 012120201120201012201012120, ...) and looking at n-th digit of a term with sufficient digits.

a(n) = A004128(n) mod 3. [Gary W. Adamson, Aug 24 2008]