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%I #10 May 25 2024 23:34:14
%S 3,3,6,11,18,30,52,91,159,278,487,854,1498,2628,4611,8091,14198,24915,
%T 43722,76726,134644,236283,414647,727654,1276943,2240878,3932466,
%U 6900996,12110403,21252275,37295142,65448411,114853954,201554638,353703732,620706779
%N Number of maximal independent vertex sets and minimal vertex covers in the n-trapezohedral graph.
%C Extended to n=1 using the formula/recurrence.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MaximalIndependentVertexSet.html">Maximal Independent Vertex Set</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MinimalVertexCover.html">Minimal Vertex Cover</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TrapezohedralGraph.html">Trapezohedral Graph</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,2,-1).
%F a(n) = A001608(2n) + 1.
%F a(n) = A109377(n+2) + 1.
%F a(n) = 3*a(n-1) - 3*a(n-2) + 2*a(n-3) - a(n-4).
%F G.f.: x*(3-6*x+6*x^2-4*x^3)/(1-3*x+3*x^2-2*x^3+x^4).
%t Table[1 + RootSum[-1 + # - 2 #^2 + #^3 &, #^n &], {n, 0, 20}]
%t LinearRecurrence[{3, -3, 2, -1}, {3, 3, 6, 11}, 20]
%t CoefficientList[Series[(3 - 6 x + 6 x^2 - 4 x^3)/(1 - 3 x + 3 x^2 - 2 x^3 + x^4), {x, 0, 20}], x]
%Y Cf. A001608, A109377.
%K nonn,easy
%O 1,1
%A _Eric W. Weisstein_, Nov 25 2023