A010872 a(n) = n mod 3.

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

Offset: 0

Author

N. J. A. Sloane

Keywords

• easy
• nonn

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

Examples

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

Links

• 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

Crossrefs

Cf. A000035, A002264, A002265, A002266, A004526, A010873, A080425, A102283.

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

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

Programs

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

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 *)
PadRight[{},120,{0,1,2}] (* or *) Mod[Range[0,120],3] (* Harvey P. Dale, Jul 20 2025 *)

PARI

my(x='x+O('x^200)); concat(0, Vec((2*x^2+x)/(1-x^3))) \\ Altug Alkan, Mar 23 2016

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

From Nicolas Bělohoubek, May 26 2025: (Start)

a(n) = (3*a(n-1)+1)*(2-a(n-1))/2 for n > 0.

a(n) = (2*a(n-1)-4)/(3*a(n-1)-4) for n > 0. (End)

Extensions

Edited by Joerg Arndt, Apr 21 2014