A053838 - OEIS

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

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OFFSET

0,3

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

Equals A004128 mod 3. [Gary W. Adamson, Aug 24 2008]

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, 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

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.

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 *)

PROG

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

CROSSREFS

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