|
| |
|
|
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
|
| |
|
|
