A053839 - OEIS

A053839

a(n) = (sum of digits of n written in base 4) modulo 4.

0, 1, 2, 3, 1, 2, 3, 0, 2, 3, 0, 1, 3, 0, 1, 2, 1, 2, 3, 0, 2, 3, 0, 1, 3, 0, 1, 2, 0, 1, 2, 3, 2, 3, 0, 1, 3, 0, 1, 2, 0, 1, 2, 3, 1, 2, 3, 0, 3, 0, 1, 2, 0, 1, 2, 3, 1, 2, 3, 0, 2, 3, 0, 1, 1, 2, 3, 0, 2, 3, 0, 1, 3, 0, 1, 2, 0, 1, 2, 3, 2, 3, 0, 1, 3, 0, 1, 2, 0, 1, 2, 3, 1, 2, 3, 0, 3, 0, 1, 2, 0, 1, 2, 3, 1

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OFFSET

0,3

COMMENTS

a(n) is the third row of the array in A141803. - Andrey Zabolotskiy, May 16 2016

This is the fixed point of the morphism 0->0123, 1->1230, 2->2301, 3->3012 starting with 0. Let t be the (nonperiodic) sequence of positions of 0, and likewise, u for 1, v for 2, and w for 3; then t(n)/n -> 4, u(n)/n -> 4, v(n)/n -> 4, w(n)/n -> 4, and t(n) + u(n) + v(n) + w(n) = 16*n - 6 for n >= 1. - Clark Kimberling, May 31 2017

LINKS

Robert Israel, Table of n, a(n) for n = 0..10000

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.

Robert Walker, Self Similar Sloth Canon Number Sequences

Index entries for sequences that are fixed points of mappings

FORMULA

a(n) = A010873(A053737(n)). - Andrey Zabolotskiy, May 18 2016

G.f. G(x) satisfies x^81*G(x) - (x^72+x^75+x^78+x^81)*G(x^4) + (x^48+x^60+x^63-x^64+x^72+x^75-x^76+x^78-x^79-x^88-x^91-x^94)*G(x^16) + (-1+x^16-x^48-x^60-x^63+2*x^64+x^76+x^79-x^80+x^112+x^124+x^127-x^128-x^140-x^143)*G(x^64) + (1-x^16-x^64+x^80-x^256+x^272+x^320-x^336)*G(x^256) = 0. - Robert Israel, May 18 2016

EXAMPLE

First three iterations of the morphism 0->0123, 1->1230, 2->2301, 3->3012:

0123

0123123023013012

0123123023013012123023013012012323013012012312303012012312302301

MAPLE

seq(convert(convert(n, base, 4), `+`) mod 4, n=0..100); # Robert Israel, May 18 2016

MATHEMATICA

Mod[Total@ IntegerDigits[#, 4], 4] & /@ Range[0, 120] (* Michael De Vlieger, May 17 2016 *)

s = Nest[Flatten[# /. {0 -> {0, 1, 2, 3}, 1 -> {1, 2, 3, 0}, 2 -> {2, 3, 0, 1}, 3 -> {3, 0, 1, 2}}] &, {0}, 9]; (* - Clark Kimberling, May 31 2017 *)

PROG

(PARI) a(n) = vecsum(digits(n, 4)) % 4; \\ Michel Marcus, May 16 2016