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A079978 Characteristic function of multiples of three. 41
1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

Period 3: repeat [1, 0, 0].

a(n)=1 if n=3k, a(n)=0 otherwise.

Decimal expansion of 1/999.

Number of permutations satisfying -k<=p(i)-i<=r and p(i)-i not in I, i=1..n, with k=1, r=2, I={0,1}.

a(n) is also the number of partitions of n with every part being three (a(0)=1 because the empty partition has no parts). Hence a(n) is also the number of 2-regular graphs on n vertices with each component having girth 3. - Jason Kimberley, Oct 02 2011

Euler transformation of A185013. - Jason Kimberley, Oct 02 2011

REFERENCES

D. H. Lehmer, Permutations with strongly restricted displacements. Combinatorial theory and its applications, II (Proc. Colloq., Balatonfured, 1969), pp. 755-770. North-Holland, Amsterdam, 1970.

LINKS

Table of n, a(n) for n=0..86.

Vladimir Baltic, On the number of certain types of strongly restricted permutations, Applicable Analysis and Discrete Mathematics Vol. 4, No 1 (2010), 119-135

N. Gogin and A. Mylläri, Padovan-like sequences and Bell polynomials, Proceedings of Applications of Computer Algebra ACA, 2013.

Index entries for characteristic functions

Index entries for linear recurrences with constant coefficients, signature (0,0,1).

FORMULA

a(n) = a(n-3) for n>2.

G.f.: 1/(1-x^3).

a(n) = (1+e^(i*Pi*A002487(n)))/2, i=sqrt(-1). - Paul Barry, Jan 14 2005

a(n) = floor(cos(n*(2/3)* Pi) + 1/2). - Paolo P. Lava, Aug 22 2006

a(n) = -1*((1 - cos(n*(2/3)* Pi))/(3/2) - 1). - Cédric Christian Bernard Gagneux, Feb 04 2014

Additive with a(p^e) = 1 if p = 3, 0 otherwise.

a(n) = -1*((n^2 mod 3)-1) - Paolo P. Lava, Oct 02 2006

a(n) = ((n+1) mod 3) mod 2. Also: a(n) = 1/2*(1+(-1)^(n+floor((n+1)/3))). - Hieronymus Fischer, May 29 2007

a(n) = 1 - A011655(n). - Reinhard Zumkeller, Nov 30 2009

a(n) = (1+(-1)^(2*n/3)+(-1)^(-2*n/3))/3. - Jaume Oliver Lafont, May 13 2010

For the general case: the characteristic function of numbers that are multiples of m is a(n) = floor(n/m) - floor((n-1)/m), m,n > 0. - Boris Putievskiy, May 08 2013

a(n) = floor( ((n-1) mod 3)/2 ). - Wesley Ivan Hurt, Jun 29 2013

a(n) = (2^(n mod 3)) mod 2. - Olivier Gérard, Jul 04 2013

a(n) = ((w^(2*n))+(w^n)+1)/3, w = (-1+i*sqrt(3))/2 (w is a primitive 3rd root of unity). - Bogart B. Strauss, Jul 20 2013

E.g.f.: (exp(x) + 2*exp(-x/2)*cos(sqrt(3)*x/2))/3. - Geoffrey Critzer, Nov 03 2014

a(n) = (sin(Pi*(n+1)/3)^2)*(2/3) + sin(Pi*(n+1)*2/3)*(1/sqrt(3)). - Mikael Aaltonen, Jan 03 2015

a(n) = (2*n^2 + 1) mod 3. The characteristic function of numbers that are multiples of 2k+1 is (2*k*n^(2*k) + 1) mod (2k+1). Example: A058331(n) mod 3 for k=1, A211412(n) mod 5 for k=2, ... - Eric Desbiaux, Dec 25 2015

a(n) = floor(2*(n-1)/3) - 2*floor((n-1)/3). - Wesley Ivan Hurt, Jul 25 2016

MAPLE

seq(op([1, 0, 0]), n=0..50); # Wesley Ivan Hurt, Jun 30 2016

MATHEMATICA

Table[Boole[IntegerQ[n/3]], {n, 0, 127}] (* Michael De Vlieger, Jan 03 2015, after Alonso del Arte at A121262 *)

PROG

(PARI) a(n)=!(n%3) \\ Jaume Oliver Lafont, Mar 01 2009

(Haskell)

a079978 = fromEnum . (== 0) . (`mod` 3)

a079978_list = cycle [1, 0, 0]

-- Reinhard Zumkeller, Aug 28 2012, Nov 26 2011

(MAGMA) &cat[[1, 0, 0]^^30]; // Vincenzo Librandi, Dec 26 2015

CROSSREFS

Cf. A002524-A002529, A072827, A072850-A072856, A079955-A080014, A022003.

Essentially the same as A022003.

Partial sums are given by A002264(n+3).

Characteristic function of multiples of g: A000007 (g=0), A000012 (g=1), A059841 (g=2), this sequence (g=3), A121262 (g=4), A079998 (g=5), A079979 (g=6), A082784 (g=7). - Jason Kimberley, Oct 14 2011

Sequence in context: A037011 A070563 A024692 * A164704 A245485 A068429

Adjacent sequences:  A079975 A079976 A079977 * A079979 A079980 A079981

KEYWORD

nonn,easy

AUTHOR

Vladimir Baltic, Feb 17 2003

EXTENSIONS

Name simplified by Ralf Stephan, Nov 22 2010

Name changed by Jason Kimberley, Oct 14 2011

STATUS

approved

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Last modified May 29 12:59 EDT 2017. Contains 287247 sequences.