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A304567 Number of maximum irredundant sets in the n-antiprism graph.

%I A304567 #21 Feb 16 2025 08:33:54
%S A304567 1,4,15,28,20,15,161,48,15,540,88,15,1365,140,15,2896,204,15,5453,280,
%T A304567 15,9416,368,15,15225,468,15,23380,580,15,34441,704,15,49028,840,15,
%U A304567 67821,988,15,91560,1148,15,121045,1320,15,157136,1504,15,200753,1700
%N A304567 Number of maximum irredundant sets in the n-antiprism graph.
%C A304567 Sequence extrapolated to n=1 using formula. - _Andrew Howroyd_, May 20 2018
%H A304567 Andrew Howroyd, <a href="/A304567/b304567.txt">Table of n, a(n) for n = 1..1000</a>
%H A304567 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/AntiprismGraph.html">Antiprism Graph</a>
%H A304567 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MaximumIrredundantSet.html">Maximum Irredundant Vertex Set</a>
%H A304567 <a href="/index/Rec#order_15">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,5,0,0,-10,0,0,10,0,0,-5,0,0,1).
%F A304567 From _Andrew Howroyd_, May 21 2018: (Start)
%F A304567 a(n) = 5*a(n-3) - 10*a(n-6) + 10*a(n-9) - 5*a(n-12) + a(n-15) for n > 15.
%F A304567 a(3*k) = 15, a(3*k+1) = (3*k+1)*(k+1)*(5*k^2+10*k+6)/6, a(3*k+2) = 2*(k+1)*(3*k+2). (End)
%F A304567 G.f.: x*(1 + 4*x + 15*x^2 + 23*x^3 - 60*x^5 + 31*x^6 - 12*x^7 + 90*x^8 + 5*x^9 + 8*x^10 - 60*x^11 + 15*x^14) / ((1 - x)^5*(1 + x + x^2)^5). - _Colin Barker_, May 22 2018
%t A304567 Table[Piecewise[{{15, Mod[n, 3] == 0}, {n (n + 2) (29 + 20 n + 5 n^2)/162, Mod[n, 3] == 1}, {2 n (n + 1)/3, Mod[n, 3] == 2}}], {n, 20}]
%t A304567 LinearRecurrence[{0, 0, 5, 0, 0, -10, 0, 0, 10, 0, 0, -5, 0, 0, 1}, {1, 4, 15, 28, 20, 15, 161, 48, 15, 540, 88, 15, 1365, 140, 15}, 20]
%t A304567 Table[(2430 + 166 n + 177 n^2 + 30 n^3 + 5 n^4 - (-4860 + 166 n + 177 n^2 + 30 n^3 + 5 n^4) Cos[2 n Pi/3] + Sqrt[3] n (-50 - 39 n + 30 n^2 + 5 n^3) Sin[2 n Pi/3])/486, {n, 20}]
%t A304567 CoefficientList[Series[15/(1 - x^3) x^2 - (1 + 23 x^3 + 31 x^6 + 5 x^9)/(-1 + x^3)^5 - (4 x (1 + 2 x^3))/(-1 + x^3)^3, {x, 0, 20}], x]
%o A304567 (PARI) a(n)={if(n%3==0, 15, my(k=n\3); n*(k+1)*if(n%3==1, (5*k^2+10*k+6)/6, 2))} \\ _Andrew Howroyd_, May 20 2018
%o A304567 (PARI) Vec(x*(1 + 4*x + 15*x^2 + 23*x^3 - 60*x^5 + 31*x^6 - 12*x^7 + 90*x^8 + 5*x^9 + 8*x^10 - 60*x^11 + 15*x^14) / ((1 - x)^5*(1 + x + x^2)^5) + O(x^50)) \\ _Colin Barker_, May 22 2018
%Y A304567 Cf. A290510, A291054.
%K A304567 nonn,easy
%O A304567 1,2
%A A304567 _Eric W. Weisstein_, May 14 2018
%E A304567 a(1)-a(2) and terms a(14) and beyond from _Andrew Howroyd_, May 20 2018