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
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
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
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
Comments