A300738 - OEIS

%I #31 May 04 2023 17:33:48

%S 0,0,3,4,5,9,7,4,12,25,22,25,39,49,68,100,119,144,209,289,367,484,644,

%T 841,1130,1521,1983,2601,3480,4624,6107,8100,10717,14161,18807,24964,

%U 33004,43681,57918,76729,101639,134689,178364,236196,313007,414736,549289

%N Number of minimal total dominating sets in the n-cycle graph.

%H Andrew Howroyd, <a href="/A300738/b300738.txt">Table of n, a(n) for n = 1..1000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CycleGraph.html">Cycle Graph</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TotalDominatingSet.html">Total Dominating Set</a>

%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,1,1,1,1,0,-1,-1).

%F a(n) = a(n-3) + a(n-4) + a(n-5) + a(n-6) - a(n-8) - a(n-9) for n > 9.

%F G.f.: x^3*(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)).

%F a(2*n) = A001608(n)^2.

%F a(2*n-1) = A001608(2*n-1), where A001608 are the Perrin numbers.

%t Table[RootSum[-1 - # + #^3 &, #^n &] + (1 + (-1)^n) RootSum[-1 + #^2 + #^3 &, #^(n/2) &], {n, 20}]

%t Perrin[n_] := RootSum[-1 - # + #^3 &, #^n &]; Table[With[{b = Mod[n, 2, 1]}, Perrin[n/b]^b], {n, 20}]

%t LinearRecurrence[{0, 0, 1, 1, 1, 1, 0, -1, -1}, {0, 0, 3, 4, 5, 9, 7, 4, 12}, 20]

%t CoefficientList[Series[x^2 (3 + 4 x + 5 x^2 + 6 x^3 - 8 x^5 - 9 x^6)/(1 - x^3 - x^4 - x^5 - x^6 + x^8 + x^9), {x, 0, 20}], x]

%o (PARI) 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)))

%Y Cf. A001608, A001638 (total dominating sets), A253413, A302653, A302655, A302918.

%K nonn,easy

%O 1,3

%A _Andrew Howroyd_, Apr 15 2018