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A079979 Characteristic function of multiples of six. 20


%S 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,

%T 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,

%U 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

%N Characteristic function of multiples of six.

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

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

%C Decimal expansion of 1/999999.

%C 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}.

%C 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}.

%C 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

%C This sequence is the Euler transformation of A185016. - _Jason Kimberley_, Oct 10 2011

%D 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.

%H Antti Karttunen, <a href="/A079979/b079979.txt">Table of n, a(n) for n = 0..65538</a>

%H Vladimir Baltic, <a href="http://pefmath.etf.rs/vol4num1/AADM-Vol4-No1-119-135.pdf">On the number of certain types of strongly restricted permutations</a>, Applicable Analysis and Discrete Mathematics Vol. 4, No 1 (April, 2010), 119-135

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,0,0,1).

%H <a href="/index/Ch#char_fns">Index entries for characteristic functions</a>

%F Recurrence: a(n) = a(n-6).

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

%F a(n) = (1/3)*[cos(n*(2/3)* Pi)+1/2]*[1+(-1)^n] with n>=0. - _Paolo P. Lava_, Aug 23 2006

%F This formula can be used to produce any periodic sequence of 6 numbers b,c,d,e,f,g: a(n)= b*(1/3)*[cos(n*(2/3)* Pi)+ 1/2]*[1+(-1)^n]+ c*(1/3)*[cos((n+5)*(2/3)* Pi)+ 1/2]*[1+(-1)^(n+5)]+ d*(1/3)*[cos((n+4)*(2/3)* Pi)+ 1/2]*[1+(-1)^(n+4)]+ e*(1/3)*[cos((n+3)*(2/3)* Pi)+ 1/2]*[1+(-1)^(n+3)]+ f*(1/3)*[cos((n+2)*(2/3)* Pi)+ 1/2]*[1+(-1)^(n+2)]+ g*(1/3)*[cos((n+1)*(2/3)* Pi)+ 1/2]*[1+(-1)^(n+1)]. - _Paolo P. Lava_, Aug 23 2006

%F a(n) = floor(1/2*cos(n*Pi/3)+1/2). - _Gary Detlefs_, May 16 2011

%F a(n) = floor(n/6)-floor((n-1)/6). - _Tani Akinari_, Oct 23 2012

%F 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

%F E.g.f.: (2*cos(sqrt(3)*x/2)*cosh(x/2) + cosh(x))/3. - _Vaclav Kotesovec_, Feb 15 2015

%t PadRight[{},120,{1,0,0,0,0,0}] (* _Harvey P. Dale_, Feb 19 2013 *)

%o (MAGMA) &cat[[1,0^^5]^^30];

%o (MAGMA) A079979 := func<n|IsDivisibleBy(n,6)select 1 else 0>; [A079979:n in [0..59]]; // _Jason Kimberley_, Oct 10 2011

%o (PARI) a(n)=!(n%6) \\ _Charles R Greathouse IV_, Oct 10 2011

%o (Scheme) (define (A079979 n) (if (zero? (modulo n 6)) 1 0)) ;; _Antti Karttunen_, Dec 22 2017

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

%Y 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).

%K nonn,easy

%O 0,1

%A _Vladimir Baltic_, Feb 17 2003

%E More terms from _Antti Karttunen_, Dec 22 2017

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Last modified January 23 16:44 EST 2020. Contains 331172 sequences. (Running on oeis4.)