A302650 Number of minimal total dominating sets in the n-barbell graph.

1, 1, 6, 28, 85, 201, 406, 736, 1233, 1945, 2926, 4236, 5941, 8113, 10830, 14176, 18241, 23121, 28918, 35740, 43701, 52921, 63526, 75648, 89425, 105001, 122526, 142156, 164053, 188385, 215326, 245056, 277761, 313633, 352870, 395676, 442261, 492841, 547638, 606880

Offset

1,3

Comments

Extended to a(1)-a(2) using the formula/recurrence.

The minimal total dominating sets have size either 2 or 4. There is only one minimum total dominating set, which consists of the two vertices of the connecting 2-path. - Andrew Howroyd, Jun 12 2025

Links

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Eric Weisstein's World of Mathematics, Barbell Graph.

Eric Weisstein's World of Mathematics, Minimal Total Dominating Set.

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Formula

a(n) = (2 + (n - 3)*n*(n + 1)/4)*n.

G.f.: x*(-1 + 4*x - 11*x^2 + 2*x^3)/(-1 + x)^5.

a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).

E.g.f.: exp(x)*x*(4 - 2*x + 4*x^2 + x^3)/4. - Stefano Spezia, Sep 06 2023

a(n) = binomial(n,2)^2 - (n-1)^2 + 1. - Andrew Howroyd, Jun 12 2025

Mathematica

Table[(2 + (n - 3) n (n + 1)/4) n, {n, 20}]
LinearRecurrence[{5, -10, 10, -5, 1}, {1, 1, 6, 28, 85}, 20]
CoefficientList[Series[(-1 + 4 x - 11 x^2 + 2 x^3)/(-1 + x)^5, {x, 0, 20}], x]

Prog

(Magma) [(2 + (n - 3)*n*(n + 1)/4)*n : n in [1..50]]; // Wesley Ivan Hurt, Apr 25 2023

Crossrefs

Cf. A302761.

Sequence in context: A300906 A394719 A222198 * A055711 A141255 A091321

Adjacent sequences: A302647 A302648 A302649 * A302651 A302652 A302653

Keyword

nonn,easy

Author

Eric W. Weisstein, Apr 11 2018

Status

approved