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A079979 Characteristic function of multiples of six.

%I A079979 #84 Dec 18 2023 10:03:22
%S A079979 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 A079979 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 A079979 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 A079979 Characteristic function of multiples of six.
%C A079979 Period 6: repeat [1, 0, 0, 0, 0, 0].
%C A079979 a(n)=1 if n=6k, a(n)=0 otherwise.
%C A079979 Decimal expansion of 1/999999.
%C A079979 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 A079979 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 A079979 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 A079979 This sequence is the Euler transformation of A185016. - _Jason Kimberley_, Oct 10 2011
%D A079979 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 A079979 Antti Karttunen, <a href="/A079979/b079979.txt">Table of n, a(n) for n = 0..65538</a>
%H A079979 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 A079979 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,0,0,1).
%H A079979 <a href="/index/Ch#char_fns">Index entries for characteristic functions</a>
%F A079979 a(n) = a(n-6).
%F A079979 G.f.: 1/(1-x^6).
%F A079979 a(n) = floor((1/2)*cos(n*Pi/3) + 1/2). - _Gary Detlefs_, May 16 2011
%F A079979 a(n) = floor(n/6) - floor((n-1)/6). - _Tani Akinari_, Oct 23 2012
%F A079979 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 A079979 E.g.f.: (2*cos(sqrt(3)*x/2)*cosh(x/2) + cosh(x))/3. - _Vaclav Kotesovec_, Feb 15 2015
%t A079979 PadRight[{},120,{1,0,0,0,0,0}] (* _Harvey P. Dale_, Feb 19 2013 *)
%o A079979 (Magma) &cat[[1,0^^5]^^30];
%o A079979 (Magma) A079979 := func<n|IsDivisibleBy(n,6)select 1 else 0>; [A079979:n in [0..59]];  // _Jason Kimberley_, Oct 10 2011
%o A079979 (PARI) a(n)=!(n%6) \\ _Charles R Greathouse IV_, Oct 10 2011
%o A079979 (Scheme) (define (A079979 n) (if (zero? (modulo n 6)) 1 0)) ;; _Antti Karttunen_, Dec 22 2017
%Y A079979 Cf. A002524-A002529, A010875, A072827, A072850-A072856, A079955-A080014, A097325, A122841.
%Y A079979 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 A079979 nonn,easy
%O A079979 0,1
%A A079979 _Vladimir Baltic_, Feb 17 2003
%E A079979 More terms from _Antti Karttunen_, Dec 22 2017