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A179058
Number of non-attacking placements of 3 rooks on an n X n board.
9
0, 0, 6, 96, 600, 2400, 7350, 18816, 42336, 86400, 163350, 290400, 490776, 794976, 1242150, 1881600, 2774400, 3995136, 5633766, 7797600, 10613400, 14229600, 18818646, 24579456, 31740000, 40560000, 51333750, 64393056, 80110296
OFFSET
1,3
COMMENTS
Also the number of 3-cycles in the n X n rook complement graph. - Eric W. Weisstein, Sep 05 2017
Also the number of 6-cycles in the complete tripartite graph K_n,n,n. - Eric W. Weisstein, Dec 07 2023
Essentially the same as A303212. - Eric W. Weisstein, Dec 06 2023
LINKS
Seth Chaiken, Christopher R. H. Hanusa and Thomas Zaslavsky, A q-Queens Problem. IV. Queens, Bishops, Nightriders (and Rooks), arXiv preprint arXiv:1609.00853 [math.CO], 2016-2020.
Eric Weisstein's World of Mathematics, Complete Tripartite Graph.
Eric Weisstein's World of Mathematics, Graph Cycle.
Eric Weisstein's World of Mathematics, Rook Complement Graph.
Eric Weisstein's World of Mathematics, Rook Graph.
FORMULA
a(n) = 3!*binomial(n, 3)^2.
a(n) = (n^2*(2-3*n+n^2)^2)/6. - Colin Barker, Jan 08 2013
G.f.: -6*x^3*(x+1)*(x^2+8*x+1) / (x-1)^7. - Colin Barker, Jan 08 2013
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
From Amiram Eldar, Nov 02 2021: (Start)
Sum_{n>=3} 1/a(n) = 3*Pi^2/2 - 117/8.
Sum_{n>=3} (-1)^(n+1)/a(n) = 21/8 - Pi^2/4. (End)
MATHEMATICA
(* Start from Eric W. Weisstein, Sep 05 2017 *)
Table[3! Binomial[n, 3]^2, {n, 20}]
3! Binomial[Range[20], 3]^2
LinearRecurrence[{7, -21, 35, -35, 21, -7, 1}, {0, 0, 6, 96, 600, 2400, 7350}, 20]
CoefficientList[Series[-((6 x^2 (1 + 9 x + 9 x^2 + x^3))/(-1 + x)^7), {x, 0, 20}], x]
(* End *)
a[n_] := If[n<3, 0, Coefficient[n!*LaguerreL[n, x], x, n-3] // Abs];
Array[a, 30] (* Jean-François Alcover, Jun 14 2018, after A144084 *)
PROG
(PARI) a(n) = 3!*binomial(n, 3)^2; \\ Andrew Howroyd, Feb 13 2018
CROSSREFS
Column k=3 of A144084.
Cf. A163102 (2 rooks), A179059 (4 rooks).
Cf. A291910 (4-cycles), A291911 (5-cycles), A291912 (6-cycles).
Sequence in context: A275086 A222971 A196813 * A303212 A226549 A053338
KEYWORD
easy,nonn
AUTHOR
Thomas Zaslavsky, Jun 27 2010
STATUS
approved