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%I #21 Sep 08 2022 08:46:19
%S 24,77,178,373,724,1331,2364,4127,7186,12625,22558,41153,76680,145607,
%T 280792,547867,1078006,2133461,4238634,8442221,16841500,33630907,
%U 67199188,134323703,268559034,537014201,1073907094,2147673337,4295184016,8590181135
%N Number of irredundant sets in the n-crown graph.
%C From _Andrew Howroyd_, Aug 11 2017: (Start)
%C For n > 3 the irredundant sets are:
%C - 2*(2^n-1): any number of vertices from one side of the graph
%C - n^2: any two vertices on opposite sides
%C - n*(n-1)*(n-2): two vertices on one side and one non-opposing on the other
%C - n*(n-1)*(n-2)*(n-3)/4: two non-opposing vertices from each side
%C - 1: the empty set
%C (End)
%H Vincenzo Librandi, <a href="/A290710/b290710.txt">Table of n, a(n) for n = 3..1000</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CrownGraph.html">Crown Graph</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/IrredundantSet.html">Irredundant Set</a>
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (7, -20, 30, -25, 11, -2).
%F a(n) = 2^(n+1) + n*(n^3 - 2*n^2 + 3*n + 2)/4 - 1 for n > 3. - _Andrew Howroyd_, Aug 11 2017
%F a(n) = 7*a(n-1) - 20*a(n-2) + 30*a(n-3) - 25*a(n-4) + 11*a(n-5) - 2*a(n-6) for n > 9.
%F G.f.: (x^3 (24 - 91 x + 119 x^2 - 53 x^3 - 37 x^4 + 44 x^5 - 12 x^6))/((-1 + x)^5 (-1 + 2 x)).
%t Table[If[n == 3, 24, 2^(n + 1) + n*(n^3 - 2 n^2 + 3 n + 2)/4 - 1], {n, 3, 20}]
%t Join[{24}, LinearRecurrence[{7, -20, 30, -25, 11, -2}, {77, 178, 373, 724, 1331, 2364}, 20]]
%t CoefficientList[Series[(24 - 91 x + 119 x^2 - 53 x^3 - 37 x^4 + 44 x^5 - 12 x^6)/((-1 + x)^5 (-1 + 2 x)), {x, 0, 20}], x]
%o (PARI) a(n)=if(n<4, [4,9,24][n], 2^(n+1) + n*(n^3 - 2*n^2 + 3*n + 2)/4 - 1); \\ _Andrew Howroyd_, Aug 11 2017
%o (Magma) [24] cat [2^(n+1)+n*(n^3-2*n^2+3*n+2)/4-1: n in [4..40]]; // _Vincenzo Librandi_, Mar 17 2018
%K nonn
%O 3,1
%A _Eric W. Weisstein_, Aug 09 2017
%E a(13)-a(32) from _Andrew Howroyd_, Aug 11 2017
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