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A053839 (Sum of digits of n written in base 4) modulo 4. 8
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

a(n) is the third row of the array in A141803, so a(n) = A141803(A034856(n+1)) for n>0. - 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

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

CROSSREFS

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

Sequence in context: A227552 A205003 A159956 * A047896 A073645 A179542

Adjacent sequences:  A053836 A053837 A053838 * A053840 A053841 A053842

KEYWORD

base,nonn

AUTHOR

Henry Bottomley, Mar 28 2000

STATUS

approved

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Last modified February 19 12:48 EST 2018. Contains 299333 sequences. (Running on oeis4.)