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%I #44 Dec 21 2021 10:15:06
%S 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,
%T 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,
%U 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
%N a(n) = (sum of digits of n written in base 4) modulo 4.
%C a(n) is the third row of the array in A141803. - _Andrey Zabolotskiy_, May 16 2016
%C 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
%H Robert Israel, <a href="/A053839/b053839.txt">Table of n, a(n) for n = 0..10000</a>
%H Glen Joyce C. Dulatre, Jamilah V. Alarcon, Vhenedict M. Florida and Daisy Ann A. Disu, <a href="http://docplayer.net/87034980-Vol-15-no-2-april-2017-dmmmsu-cas-science-monitor.html">On Fractal Sequences</a>, DMMMSU-CAS Science Monitor (2016-2017) Vol. 15 No. 2, 109-113.
%H Robert Walker, <a href="http://robertinventor.com/ftswiki/Self_Similar_Sloth_Canon_Number_Sequences">Self Similar Sloth Canon Number Sequences</a>
%H <a href="/index/Fi#FIXEDPOINTS">Index entries for sequences that are fixed points of mappings</a>
%F a(n) = A010873(A053737(n)). - _Andrey Zabolotskiy_, May 18 2016
%F 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
%e First three iterations of the morphism 0->0123, 1->1230, 2->2301, 3->3012:
%e 0123
%e 0123123023013012
%e 0123123023013012123023013012012323013012012312303012012312302301
%p seq(convert(convert(n,base,4),`+`) mod 4, n=0..100); # _Robert Israel_, May 18 2016
%t Mod[Total@ IntegerDigits[#, 4], 4] & /@ Range[0, 120] (* _Michael De Vlieger_, May 17 2016 *)
%t 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 *)
%o (PARI) a(n) = vecsum(digits(n,4)) % 4; \\ _Michel Marcus_, May 16 2016
%o (PARI) a(n) = sumdigits(n, 4) % 4; \\ _Michel Marcus_, Jul 04 2018
%Y Cf. A010060, A141803, A287552, A287553, A287554, A287555.
%K base,nonn
%O 0,3
%A _Henry Bottomley_, Mar 28 2000
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