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%I #16 Mar 04 2024 21:29:43
%S 0,0,4,36,124,320,672,1260,2152,3456,5260,7700,10884,14976,20104,
%T 26460,34192,43520,54612,67716,83020,100800,121264,144716,171384,
%U 201600,235612,273780,316372,363776,416280,474300,538144,608256,684964,768740,859932
%N Number of cycles of length 3 in the queen graph associated with an n X n chessboard.
%H Colin Barker, <a href="/A144298/b144298.txt">Table of n, a(n) for n = 0..1000</a>
%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GraphCycle.html">Graph Cycle</a>.
%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/QueenGraph.html">Queen Graph</a>.
%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (3,-1,-5,5,1,-3,1).
%F G.f.: 4*x^2*(1 + 6*x + 5*x^2 + x^3 - x^4) / ((1 - x)^5*(1 + x)^2).
%F a(n) = A030117(n) + (3*n-1)*binomial(n,3).
%t Table[n (5 - 10 n + 2 n^2 + 2 n^3 - (-1)^n)/4, {n, 20}] (* _Eric W. Weisstein_, Jun 19 2017 *)
%t LinearRecurrence[{3, -1, -5, 5, 1, -3, 1}, {0, 4, 36, 124, 320, 672, 1260}, 20] (* _Eric W. Weisstein_, Jun 19 2017 *)
%t CoefficientList[Series[(4 x (-1 - 6 x - 5 x^2 - x^3 + x^4))/((-1 + x)^5 (1 + x)^2), {x, 0, 20}], x] (* _Eric W. Weisstein_, Jun 19 2017 *)
%o (PARI) concat(vector(2), Vec(4*x^2*(1 + 6*x + 5*x^2 + x^3 - x^4) / ((1 - x)^5*(1 + x)^2) + O(x^30))) \\ _Colin Barker_, May 11 2017
%Y Cf. A156001 (4-cycles), A288916 (5-cycles), A288917 (6-cycles).
%K nonn,easy
%O 0,3
%A _Sergey Perepechko_, Dec 04 2008
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