A361185 - OEIS

%I #12 Apr 19 2023 02:46:39

%S 0,0,15,264,1700,6900,21315,54880,123984,253800,480975,856680,1450020,

%T 2351804,3678675,5577600,8230720,11860560,16735599,23176200,31560900,

%U 42333060,56007875,73179744,94530000,120835000,152974575,191940840,238847364,294938700

%N Number of chordless cycles in the n X n rook complement graph.

%C Using the convention that chordless cycles have length >= 4.

%C All chordless cycles in the rook complement graph have a cycle length of either 4 or 6. - _Andrew Howroyd_, Mar 03 2023

%H Andrew Howroyd, <a href="/A361185/b361185.txt">Table of n, a(n) for n = 1..1000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ChordlessCycle.html">Chordless Cycle</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RookComplementGraph.html">Rook Complement Graph</a>

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1).

%F a(n) = 2*binomial(n,2)*binomial(n,3) + 9*binomial(n,3)^2 + 12*binomial(n,4)*binomial(n,2). - _Andrew Howroyd_, Mar 03 2023

%F a(n) = (n - 2)*(n - 1)^2*n^2*(6*n - 13)/12.

%F a(n) = 7*a(n-1)-21*a(n-2)+35*a(n-3)-35*a(n-4)+21*a(n-5)-7*a(n-6)+a(n-7).

%F G.f.: x^3*(15+159*x+167*x^2+19*x^3)/(1-x)^7.

%t Table[(n - 2) (n - 1)^2 n^2 (6 n - 13)/12, {n, 20}]

%t LinearRecurrence[{7, -21, 35, -35, 21, -7, 1}, {0, 0, 15, 264, 1700, 6900, 21315}, 20]

%t CoefficientList[Series[x^2 (15 + 159 x + 167 x^2 + 19 x^3)/(1 - x)^7, {x, 0, 20}], x]

%o (PARI) a(n) = 2*binomial(n,2)*binomial(n,3) + 9*binomial(n,3)^2 + 12*binomial(n,4)*binomial(n,2) \\ _Andrew Howroyd_, Mar 03 2023

%Y Cf. A070968, A360854.

%K nonn

%O 1,3

%A _Eric W. Weisstein_, Mar 03 2023

%E Terms a(8) and beyond from _Andrew Howroyd_, Mar 03 2023