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

1,6

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, Total Dominating Set

Formula

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

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 *)

Prog

(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))))

Crossrefs

Cf. A300738.

Sequence in context: A330334 A087741 A054991 * A316149 A047071 A124287

Adjacent sequences: A302915 A302916 A302917 * A302919 A302920 A302921

Keyword

nonn

Author

Andrew Howroyd, Apr 15 2018

Status

approved