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%I #11 Oct 06 2019 01:51:05
%S 0,0,1,1,1,2,1,1,2,3,2,4,3,4,6,7,7,10,11,17,19,23,28,38,46,60,75,96,
%T 120,160,197,257,327,420,539,701,892,1155,1488,1928,2479,3220,4148,
%U 5381,6961,9030,11687,15183,19673,25563,33174,43128,56010,72864,94719
%N Number of nonequivalent minimal total dominating sets in the n-cycle graph up to rotation.
%H Andrew Howroyd, <a href="/A302918/b302918.txt">Table of n, a(n) for n = 1..1000</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CycleGraph.html">Cycle Graph</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TotalDominatingSet.html">Total Dominating Set</a>
%F a(n) = (1/n) * Sum_{d|n} phi(n/d) * A300738(d).
%t A300738 = DifferenceRoot[Function[{f, n}, {f[n] + f[n+1] - f[n+3] - f[n+4] - f[n+5] - f[n+6] + f[n+9] == 0, f[1]==0, f[2]==0, f[3]==3, f[4]==4, f[5]==5, f[6]==9, f[7]==7, f[8]==4, f[9]==12}]];
%t a[n_] := (1/n) Sum[EulerPhi[n/d] A300738[d], {d, Divisors[n]}];
%t a /@ Range[1, 55] (* _Jean-François Alcover_, Sep 21 2019 *)
%o (PARI)
%o NecklaceT(v)={vector(#v, n, sumdiv(n,d,eulerphi(n/d)*v[d])/n)}
%o NecklaceT(concat([0,0],Vec((3 + 4*x + 5*x^2 + 6*x^3 - 8*x^5 - 9*x^6)/((1 - x^2 - x^3)*(1 + x^2 - x^6)) + O(x^50))))
%Y Cf. A300738.
%K nonn
%O 1,6
%A _Andrew Howroyd_, Apr 15 2018
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