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A010872
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a(n) = n mod 3.
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94
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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, 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, 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
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OFFSET
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0,3
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COMMENTS
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Fixed point of morphism 0 -> 01, 1 -> 20, 2 -> 12.
Complement of A002264, since 3*A002264(n) + a(n) = n. - Hieronymus Fischer, Jun 01 2007
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LINKS
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Table of n, a(n) for n=0..104.
Ralph E. Griswold, Shaft Sequences
Index entries for linear recurrences with constant coefficients, signature (0,0,1).
Index entries for sequences that are fixed points of mappings
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FORMULA
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a(n) = n - 3*floor(n/3) = a(n-3).
G.f.: (2*x^2+x)/(1-x^3). - Mario Catalani (mario.catalani(AT)unito.it), Jan 08 2003
a(n) = 1 + (1-2*cos(2*Pi*(n-1)/3)) * sin(2*Pi*(n-1)/3)) / sqrt(3).
Complex representation: a(n) = (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, May 29 2007; corrected by Guenther Schrack, Sep 23 2019
a(n) = (16/9)*((sin(Pi*(n-2)/3))^2+2*(sin(Pi*(n-1)/3))^2)*(sin(Pi*n/3))^2.
a(n) = (4/3)*(|sin(Pi*(n-2)/3)|+2*|sin(Pi*(n-1)/3)|)*|sin(Pi*n/3)|.
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 periodic sequence. - Hieronymus Fischer, Jun 01 2007
Trigonometric formulas above edited for better readability by Hieronymus Fischer, Nov 22 2011
a(n) = 3 - a(n-1) - a(n-2) for n > 1. - Reinhard Zumkeller, Apr 13 2008
a(n) = 1-2*sin(4*Pi*(n+2)/3)/sqrt(3). - Jaume Oliver Lafont, Dec 05 2008
From Wesley Ivan Hurt, May 27 2015, Mar 22 2016: (Start)
a(n) = 1 - 0^((-1)^(n/3)-(-1)^n) + 0^((-1)^((n+1)/3)+(-1)^n).
a(n) = 1 + (-1)^((2*n+4)/3)/3 + (-1)^((-2*n-4)/3)/3 + 2*(-1)^((2*n+2)/3)/3 + 2*(-1)^((-2*n-2)/3)/3.
a(n) = 1 + 2*cos(Pi*(2*n+4)/3)/3 + 4*cos(Pi*(2*n+2)/3)/3. (End)
a(n) = (r^n*(r-1) - r^(2*n)*(r + 2) + 3)/3 where r = (-1 + sqrt(-3))/2. - Guenther Schrack, Sep 23 2019
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EXAMPLE
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G.f. = x + 2*x^2 + x^4 + 2*x^5 + x^7 + 2*x^8 + x^10 + 2*x^11 + x^13 + ...
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MAPLE
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A010872:=n->(n mod 3): seq(A010872(n), n=0..100); # Wesley Ivan Hurt, May 27 2015
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MATHEMATICA
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Nest[ Function[ l, {Flatten[(l /. {0 -> {0, 1}, 1 -> {2, 0}, 2 -> {1, 2}})]}], {0}, 7] (* Robert G. Wilson v, Feb 28 2005 *)
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PROG
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(Haskell)
a010872 = (`mod` 3)
a010872_list = cycle [0, 1, 2] -- Reinhard Zumkeller, May 26 2012
(MAGMA) [n mod 3 : n in [0..100]]; // Wesley Ivan Hurt, May 27 2015
(PARI) x='x+O('x^200); concat(0, Vec((2*x^2+x)/(1-x^3))) \\ Altug Alkan, Mar 23 2016
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CROSSREFS
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Cf. A000035, A010873. A080425, A004526, A002264, A002265, A002266.
Cf. partial sums: A130481.
Other related sequences are A130482, A130483, A130484, A130485.
Sequence in context: A112248 A244860 A308009 * A220663 A220659 A025858
Adjacent sequences: A010869 A010870 A010871 * A010873 A010874 A010875
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KEYWORD
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easy,nonn
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AUTHOR
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N. J. A. Sloane
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EXTENSIONS
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Edited by Joerg Arndt, Apr 21 2014
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STATUS
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approved
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