A302918 Number of nonequivalent minimal total dominating sets in the n-cycle graph up to rotation.

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, 120, 160, 197, 257, 327, 420, 539, 701, 892, 1155, 1488, 1928, 2479, 3220, 4148, 5381, 6961, 9030, 11687, 15183, 19673, 25563, 33174, 43128, 56010, 72864, 94719

Offset: 1

Author

Andrew Howroyd, Apr 15 2018

Keywords

• nonn

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..1000
• Eric Weisstein's World of Mathematics, Cycle Graph.
• Eric Weisstein's World of Mathematics, Minimal Total Dominating Set.

Crossrefs

Cf. A300738.

Programs

Mathematica

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}]];
a[n_] := (1/n) Sum[EulerPhi[n/d] A300738[d], {d, Divisors[n]}];
a /@ Range[1, 55] (* Jean-François Alcover, Sep 21 2019 *)

PARI

NecklaceT(v)={vector(#v, n, sumdiv(n,d,eulerphi(n/d)*v[d])/n)}
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))))

Formula

a(n) = (1/n) * Sum_{d|n} phi(n/d) * A300738(d).