A010872 - OEIS

A010872

a(n) = n mod 3.

123

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

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

Decimal expansion of 4/333. - Elmo R. Oliveira, Feb 19 2024

Period 3: repeat [0, 1, 2]. - Elmo R. Oliveira, Jun 20 2024

LINKS

Table of n, a(n) for n=0..104.

Paul Barry, On a Central Transform of Integer Sequences, arXiv:2004.04577 [math.CO], 2020.

Ralph E. Griswold, Shaft Sequences

Ralph E. Griswold, Shaft Sequences [From the Wayback machine]

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

From Hieronymus Fischer, May 29 2007: (Start)

a(n) = 1 + (1-2*cos(2*Pi*(n-1)/3)) * sin(2*Pi*(n-1)/3)) / sqrt(3).

a(n) = (1-r^n)*(1+r^n/(1-r)) where r=exp(2*Pi*i/3)=(-1+sqrt(3)*i)/2 and i=sqrt(-1). [corrected by Guenther Schrack, Sep 23 2019] (End)

From Hieronymus Fischer, Jun 01 2007: (Start)

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)). (End)

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

E.g.f.: exp(x) - exp(-x/2)*(cos(sqrt(3)*x/2) + sin(sqrt(3)*x/2)/sqrt(3)). - Stefano Spezia, Mar 01 2020

a(n) = A010882(n) - 1 = A131555(2*n) = A131555(2*n+1). - Elmo R. Oliveira, Jun 25 2024

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, A002264, A002265, A002266, A004526, A010873, A080425, A102283.

Cf. A010882, A130481 (partial sums), A131555.

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

KEYWORD

easy,nonn

AUTHOR

N. J. A. Sloane

EXTENSIONS

Edited by Joerg Arndt, Apr 21 2014

STATUS

approved