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%I #17 Aug 15 2022 08:35:17
%S 0,0,82,430,1030,1882,2986,4342,5950,7810,9922,12286,14902,17770,
%T 20890,24262,27886,31762,35890,40270,44902,49786,54922,60310,65950,
%U 71842,77986,84382,91030,97930,105082,112486,120142,128050,136210,144622,153286,162202
%N Number of 6-cycles in the n X n king graph.
%H Colin Barker, <a href="/A288920/b288920.txt">Table of n, a(n) for n = 1..1000</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GraphCycle.html">Graph Cycle</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/KingGraph.html">King Graph</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F From _Colin Barker_, Jun 19 2017: (Start)
%F G.f.: 2*x^3*(41 + 92*x - 7*x^2) / (1 - x)^3.
%F a(n) = 550 - 534*n + 126*n^2 for n > 2.
%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 4. (End)
%F E.g.f.: 2*exp(x)*(275 - 204*x + 63*x^2) - 550 - 142*x + 7*x^2. - _Stefano Spezia_, Aug 14 2022
%t Table[If[n < 3, 0, 2 (275 - 267 n + 63 n^2)], {n, 20}]
%t CoefficientList[Series[(2 x^2 (-41 - 92 x + 7 x^2))/(-1 + x)^3, {x, 0, 20}], x]
%t Join[{0, 0}, LinearRecurrence[{3, -3, 1}, {142, -14, 82}, {3, 20}]]
%o (PARI) concat(vector(2), Vec(2*x^3*(41 + 92*x - 7*x^2) / (1 - x)^3 + O(x^50))) \\ _Colin Barker_, Jun 19 2017
%o (PARI) a(n)=if(n>2, 126*n^2-534*n+550, 0) \\ _Charles R Greathouse IV_, Jun 19 2017
%Y Cf. A016742 (number of 3-cycles).
%Y Cf. A288918 (number of 4-cycles).
%Y Cf. A288919 (number of 5-cycles).
%K nonn,easy
%O 1,3
%A _Eric W. Weisstein_, Jun 19 2017
%E More terms from _Colin Barker_, Jun 19 2017
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