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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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Also the number of 3-cycles in the n X n rook complement graph. - Eric W. Weisstein, Sep 05 2017

LINKS

Andrew Howroyd, Table of n, a(n) for n = 1..200

Christopher R. H. Hanusa, T. Zaslavsky, S. Chaiken, A q-Queens Problem. IV. Queens, Bishops, Nightriders (and Rooks), arXiv preprint arXiv:1609.00853 [math.CO], 2016.

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

Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

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

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-Fran├žois 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

Adjacent sequences:  A179055 A179056 A179057 * A179059 A179060 A179061

KEYWORD

easy,nonn

AUTHOR

Thomas Zaslavsky, Jun 27 2010

STATUS

approved

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Last modified October 17 14:31 EDT 2019. Contains 328114 sequences. (Running on oeis4.)