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A010872 a(n) = n mod 3. 80
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

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

LINKS

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

FORMULA

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) = 1/3*(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, May 29 2007

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)

EXAMPLE

G.f. = x + 2*x^2 + x^4 + 2*x^5 + x^7 + 2*x^8 + x^10 + 2*x^11 + x^13 + ...

MAPLE

A010872:=n->(n mod 3): seq(A010872(n), n=0..100); # Wesley Ivan Hurt, May 27 2015

MATHEMATICA

Nest[ Function[ l, {Flatten[(l /. {0 -> {0, 1}, 1 -> {2, 0}, 2 -> {1, 2}})]}], {0}, 7] (* Robert G. Wilson v, Feb 28 2005 *)

PROG

(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

CROSSREFS

Cf. A000035, A010873. A080425, A004526, A002264, A002265, A002266.

Cf. partial sums: A130481.

Other related sequences are A130482, A130483, A130484, A130485.

Sequence in context: A134979 A112248 A244860 * A220663 A220659 A025858

Adjacent sequences:  A010869 A010870 A010871 * A010873 A010874 A010875

KEYWORD

easy,nonn

AUTHOR

N. J. A. Sloane

EXTENSIONS

Edited by Joerg Arndt, Apr 21 2014

STATUS

approved

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Last modified September 26 15:27 EDT 2017. Contains 292531 sequences.