A079979 Characteristic function of multiples of six.

1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1

Offset: 0

Author

Vladimir Baltic, Feb 17 2003

Keywords

• nonn
• easy

Comments

Period 6: repeat [1, 0, 0, 0, 0, 0].

a(n)=1 if n=6k, a(n)=0 otherwise.

Decimal expansion of 1/999999.

Number of permutations satisfying -k <= p(i)-i <= r and p(i)-i not in I, i=1..n, with k=3, r=3, I={-2,-1,0,1,2}.

Also, number of permutations satisfying -k <= p(i)-i <= r and p(i)-i not in I, i=1..n, with k=1, r=5, I={0,1,2,3,4}.

a(n) is also the number of partitions of n such that each part is six (a(0)=1 because the empty partition has no parts to test equality with six). Hence a(n) is also the number of 2-regular graphs on n vertices with each part having girth exactly six. - Jason Kimberley, Oct 10 2011

This sequence is the Euler transformation of A185016. - Jason Kimberley, Oct 10 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

• Antti Karttunen, Table of n, a(n) for n = 0..65538
• Vladimir Baltic, On the number of certain types of strongly restricted permutations, Applicable Analysis and Discrete Mathematics Vol. 4, No 1 (April, 2010), 119-135
• Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,1).
• Index entries for characteristic functions

Crossrefs

Cf. A002524-A002529, A010875, A072827, A072850-A072856, A079955-A080014, A097325, A122841.

Characteristic function of multiples of g: A000007 (g=0), A000012 (g=1), A059841 (g=2), A079978 (g=3), A121262 (g=4), A079998 (g=5), this sequence (g=6), A082784 (g=7).

Programs

Magma

&cat[[1,0^^5]^^30];

Magma

A079979 := func; [A079979:n in [0..59]];  // Jason Kimberley, Oct 10 2011

Mathematica

PadRight[{},120,{1,0,0,0,0,0}] (* Harvey P. Dale, Feb 19 2013 *)

PARI

a(n)=!(n%6) \\ Charles R Greathouse IV, Oct 10 2011

Scheme

(define (A079979 n) (if (zero? (modulo n 6)) 1 0)) ;; Antti Karttunen, Dec 22 2017

Formula

a(n) = a(n-6).

G.f.: 1/(1-x^6).

a(n) = floor((1/2)*cos(n*Pi/3) + 1/2). - Gary Detlefs, May 16 2011

a(n) = floor(n/6) - floor((n-1)/6). - Tani Akinari, Oct 23 2012

a(n) = (((((v^n - w^n)^2)*(2 - (-1)^n)*(w^(2*n) + w^n - 3))^2 - 144)^2)/20736, where w = (-1+i*sqrt(3))/2, v = (1+i*sqrt(3))/2. - Bogart B. Strauss, Sep 20 2013

E.g.f.: (2*cos(sqrt(3)*x/2)*cosh(x/2) + cosh(x))/3. - Vaclav Kotesovec, Feb 15 2015

Extensions

More terms from Antti Karttunen, Dec 22 2017