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

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
(PARI) a(n) = sumdigits(n, 4) % 4; \\ Michel Marcus, Jul 04 2018

Crossrefs

Cf. A010060, A141803, A287552, A287553, A287554, A287555.

Sequence in context: A227552 A205003 A159956 * A047896 A073645 A294180

Adjacent sequences: A053836 A053837 A053838 * A053840 A053841 A053842

Keyword

base,nonn

Author

Henry Bottomley, Mar 28 2000

Status

approved