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
Andrew Howroyd, Table of n, a(n) for n = 1..200
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.
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
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
KEYWORD
easy,nonn
AUTHOR
Thomas Zaslavsky, Jun 27 2010
STATUS
approved