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Number of 3-cycles in the n X n rook graph.
6

%I #18 Feb 25 2023 20:56:50

%S 0,0,6,32,100,240,490,896,1512,2400,3630,5280,7436,10192,13650,17920,

%T 23120,29376,36822,45600,55860,67760,81466,97152,115000,135200,157950,

%U 183456,211932,243600,278690,317440,360096,406912,458150,514080,574980,641136,712842,790400

%N Number of 3-cycles in the n X n rook graph.

%H Andrew Howroyd, <a href="/A288961/b288961.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/RookGraph.html">Rook Graph</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).

%F a(n) = 2*n*binomial(n,3).

%F a(n) = 5*a(n-1)-10*a(n-2)+10*a(n-3)-5*a(n-4)+a(n-5).

%F G.f.: (-2*x^3*(3+x))/(-1+x)^5.

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

%t Table[2 n Binomial[n, 3], {n, 20}]

%t LinearRecurrence[{5, -10, 10, -5, 1}, {0, 0, 6, 32, 100}, 20]

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

%o (PARI) a(n) = {2*n*binomial(n,3)} \\ _Andrew Howroyd_, Apr 26 2020

%Y Cf. A288962 (4-cycles), A288963 (5-cycles), A288960 (6-cycles).

%Y Main diagonal of A360855.

%K nonn,easy

%O 1,3

%A _Eric W. Weisstein_, Jun 20 2017

%E Terms a(31) and beyond from _Andrew Howroyd_, Apr 26 2020