This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A179058 #47 Feb 16 2025 08:33:12
%S A179058 0,0,6,96,600,2400,7350,18816,42336,86400,163350,290400,490776,794976,
%T A179058 1242150,1881600,2774400,3995136,5633766,7797600,10613400,14229600,
%U A179058 18818646,24579456,31740000,40560000,51333750,64393056,80110296
%N A179058 Number of non-attacking placements of 3 rooks on an n X n board.
%C A179058 Also the number of 3-cycles in the n X n rook complement graph. - _Eric W. Weisstein_, Sep 05 2017
%C A179058 Also the number of 6-cycles in the complete tripartite graph K_{n,n,n}. - _Eric W. Weisstein_, Dec 07 2023
%C A179058 Essentially the same as A303212. - _Eric W. Weisstein_, Dec 06 2023
%H A179058 Andrew Howroyd, <a href="/A179058/b179058.txt">Table of n, a(n) for n = 1..200</a>
%H A179058 Seth Chaiken, Christopher R. H. Hanusa and Thomas Zaslavsky, <a href="http://arxiv.org/abs/1609.00853">A q-Queens Problem. IV. Queens, Bishops, Nightriders (and Rooks)</a>, arXiv preprint arXiv:1609.00853 [math.CO], 2016-2020.
%H A179058 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CompleteTripartiteGraph.html">Complete Tripartite Graph</a>.
%H A179058 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GraphCycle.html">Graph Cycle</a>.
%H A179058 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RookComplementGraph.html">Rook Complement Graph</a>.
%H A179058 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RookGraph.html">Rook Graph</a>.
%H A179058 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1).
%F A179058 a(n) = 3!*binomial(n, 3)^2.
%F A179058 a(n) = (n^2*(2-3*n+n^2)^2)/6. - _Colin Barker_, Jan 08 2013
%F A179058 G.f.: -6*x^3*(x+1)*(x^2+8*x+1) / (x-1)^7. - _Colin Barker_, Jan 08 2013
%F A179058 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). - _Eric W. Weisstein_, Sep 05 2017
%F A179058 From _Amiram Eldar_, Nov 02 2021: (Start)
%F A179058 Sum_{n>=3} 1/a(n) = 3*Pi^2/2 - 117/8.
%F A179058 Sum_{n>=3} (-1)^(n+1)/a(n) = 21/8 - Pi^2/4. (End)
%t A179058 (* Start from _Eric W. Weisstein_, Sep 05 2017 *)
%t A179058 Table[3! Binomial[n, 3]^2, {n, 20}]
%t A179058 3! Binomial[Range[20], 3]^2
%t A179058 LinearRecurrence[{7, -21, 35, -35, 21, -7, 1}, {0, 0, 6, 96, 600, 2400, 7350}, 20]
%t A179058 CoefficientList[Series[-((6 x^2 (1 + 9 x + 9 x^2 + x^3))/(-1 + x)^7), {x, 0, 20}], x]
%t A179058 (* End *)
%t A179058 a[n_] := If[n<3, 0, Coefficient[n!*LaguerreL[n, x], x, n-3] // Abs];
%t A179058 Array[a, 30] (* _Jean-François Alcover_, Jun 14 2018, after A144084 *)
%o A179058 (PARI) a(n) = 3!*binomial(n, 3)^2; \\ _Andrew Howroyd_, Feb 13 2018
%Y A179058 Column k=3 of A144084.
%Y A179058 Cf. A163102 (2 rooks), A179059 (4 rooks).
%Y A179058 Cf. A291910 (4-cycles), A291911 (5-cycles), A291912 (6-cycles).
%K A179058 easy,nonn
%O A179058 1,3
%A A179058 _Thomas Zaslavsky_, Jun 27 2010