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
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).
Crossrefs
Cf. A302761.
Programs
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Magma
[(2 + (n - 3)*n*(n + 1)/4)*n : n in [1..50]]; // Wesley Ivan Hurt, Apr 25 2023
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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]
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
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