Search: id:A010872
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%I A010872
%S A010872 0,1,2,0,1,2,0,1,2,0,1,2,0,1,2,0,1,2,0,1,2,0,1,2,0,1,2,0,1,2,0,1,2,0,1,
%T A010872 2,0,1,2,0,1,2,0,1,2,0,1,2,0,1,2,0,1,2,0,1,2,0,1,2,0,1,2,0,1,2,0,1,2,0,
%U A010872 1,2,0,1,2,0,1,2,0,1,2,0,1,2,0,1,2,0,1,2,0,1,2,0,1,2,0,1,2,0,1,2,0,1,2
%N A010872 n mod 3.
%C A010872 Complement of A002264, since 3*A002264(n)+a(n)=n. - Hieronymus Fischer 
               (Hieronymus.Fischer(AT)gmx.de), Jun 01 2007
%H A010872 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A010872 Ralph E. Griswold, <a href="http://www.cs.arizona.edu/patterns/sequences.html">
               Shaft Sequences</a>
%F A010872 a(n) = n-3*floor(n/3) = a(n-3)
%F A010872 G.f.: (2x^2+x)/(1-x^3). a(n)=(1/2)(-1)^floor(2n/3)-(-1)^floor((2n-1)/
               3)-(3/2)(-1)^floor((2n+1)/3). a(n)=3*A022003(n)+A049347(n+2). - Mario 
               Catalani (mario.catalani(AT)unito.it), Jan 08 2003
%F A010872 Fixed point of morphism 0 -> 01, 1 -> 20, 2 -> 12.
%F A010872 a(n)=1+(1-2cos(2*pi*(n-1)/3))*sin(2*pi*(n-1)/3))/sqrt(3). There is also 
               a complex representation: a(n)=1/3*(1-r^n)*(1+r^n/(1-r)) where r=exp(2*pi/
               3*i)=(-1+sqrt(3)*i)/2 and i=sqrt(-1). - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), 
               May 29 2007
%F A010872 Other trigonometric representation: a(n) = (16/9)*((sin(pi*(n-2)/3))^2+2*(sin(pi*(n-1)/
               3))^2)*(sin(pi*n/3))^2. Also: a(n) = (4/3)*(|sin(pi*(n-2)/3)|+2*|sin(pi*(n-1)/
               3)|)*|sin(pi*n/3)|. Also: a(n) = (4/9)*((1-cos(2*pi*(n-2)/3))+2*(1-cos(2*pi*(n-1)/
               3)))*(1-cos(2*pi*n/3)). These formulas can be easily adapted to represent 
               any peridoc sequence. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), 
               Jun 01 2007
%F A010872 a(n) = 3 - a(n-1) - a(n-2) for n > 1. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Apr 13 2008
%F A010872 a(n)=1-2*sin(4*pi*(n+2)/3)/sqrt(3) [From Jaume Oliver Lafont (joliverlafont(AT)gmail.com), 
               Dec 05 2008]
%p A010872 seq(chrem( [n,n], [1,3] ), n=0..100);# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Mar 25 2009]
%t A010872 Nest[ Function[ l, {Flatten[(l /. {0 -> {0, 1}, 1 -> {2, 0}, 2 -> {1, 
               2}})]}], {0}, 7] (from Robert G. Wilson v Feb 28 2005)
%o A010872 (Other) sage: [power_mod(n,3,3 )for n in xrange(0, 105)] # [From Zerinvary 
               Lajos (zerinvarylajos(AT)yahoo.com), Oct 29 2009]
%Y A010872 Cf. A000035, A010873. A080425, A004526, A002264, A002265, A002266.
%Y A010872 Cf. partial sums: A130481. Other related sequences A130482, A130483, 
               A130484, A130485.
%Y A010872 Sequence in context: A166124 A134979 A112248 this_sequence A025858 A025684 
               A025678
%Y A010872 Adjacent sequences: A010869 A010870 A010871 this_sequence A010873 A010874 
               A010875
%K A010872 easy,nonn
%O A010872 0,3
%A A010872 N. J. A. Sloane (njas(AT)research.att.com).

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