A303212 -id:A303212 - cp's OEIS frontend

A179058 Number of non-attacking placements of 3 rooks on an n X n board.

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

Thomas Zaslavsky, Jun 27 2010

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

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).

Column k=3 of A144084.
Cf. A163102 (2 rooks), A179059 (4 rooks).
Cf. A291910 (4-cycles), A291911 (5-cycles), A291912 (6-cycles).

(* 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 *)

a(n) = 3!*binomial(n, 3)^2; \\ Andrew Howroyd, Feb 13 2018

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)

A303209 Number of total dominating sets in the n X n rook complement graph.

Original entry on oeis.org

0, 1, 334, 63935, 33543096, 68719407273, 562949953031502, 18446744073707484655, 2417851639229258338871776, 1267650600228229401496650964865, 2658455991569831745807614120307390270, 22300745198530623141535718272648360299110799

Offset: 1

Eric W. Weisstein, Apr 19 2018

nonn

The vertex sets which are not totally dominating are just those that are contained in the union of a single row and column. - Andrew Howroyd, Apr 20 2018

Andrew Howroyd, Table of n, a(n) for n = 1..50
Eric Weisstein's World of Mathematics, Rook Complement Graph
Eric Weisstein's World of Mathematics, Total Dominating Set

Cf. A292073, A303212, A347922.

a(n) = {2^(n^2) - 2*n*(2^n - 1) - 2*n^2*(2^(n-1)-1)^2 + n^2*(n-1)^2/2 + n^2 - 1} \\ Andrew Howroyd, Apr 20 2018

a(n) = 2^(n^2) - 2*n*(2^n - 1) - 2*n^2*(2^(n-1)-1)^2 + n^2*(n-1)^2/2 + n^2 - 1. - Andrew Howroyd, Apr 20 2018

Terms a(6) and beyond from Andrew Howroyd, Apr 20 2018

A347922 Number of minimal total dominating sets in the n X n rook complement graph.

Original entry on oeis.org

0, 1, 51, 492, 2500, 8925, 25431, 61936, 134352, 266625, 493075, 861036, 1433796, 2293837, 3546375, 5323200, 7786816, 11134881, 15604947, 21479500, 29091300, 38829021, 51143191, 66552432, 85650000, 109110625, 137697651, 172270476, 213792292, 263338125

Offset: 1

Eric W. Weisstein, Sep 19 2021

From Andrew Howroyd, Jan 19 2022: (Start)
The vertex sets which are not totally dominating are just those that are contained in the union of a single row and column. Minimal total dominating sets are:
  - any three vertices such that no two are in the same row or column,
  - two vertices in each of two rows/columns. (End)

Andrew Howroyd, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Minimal Total Dominating Set
Eric Weisstein's World of Mathematics, Rook Complement Graph
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A292074, A303209, A303212.

Table[(n - 1)^2 n^2 (5 n^2 - 11 n + 5)/12, {n, 20}] (* Eric W. Weisstein, May 11 2024 *)
LinearRecurrence[{7, -21, 35, -35, 21, -7, 1}, {0, 1, 51, 492, 2500, 8925, 25431}, 20] (* Eric W. Weisstein, May 11 2024 *)
CoefficientList[Series[-x (1 + 44 x + 156 x^2 + 92 x^3 + 7 x^4)/(-1 + x)^7, {x, 0, 20}], x] (* Eric W. Weisstein, May 11 2024 *)

a(n) = (5*n^2 - 11*n + 5)*n^2*(n-1)^2/12 \\ Andrew Howroyd, Jan 19 2022

From Andrew Howroyd, Jan 19 2022: (Start)
a(n) = 6*binomial(n,3)^2 + 2*binomial(n,2)^3 - binomial(n,2)^2.
a(n) = (5*n^2 - 11*n + 5)*n^2*(n-1)^2/12.
G.f.: x*(1 + 44*x + 156*x^2 + 92*x^3 + 7*x^4)/(1 - x)^7.
(End)

Terms a(6) and beyond from Andrew Howroyd, Jan 19 2022

cp's OEIS Frontend

A179058 Number of non-attacking placements of 3 rooks on an n X n board.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A303209 Number of total dominating sets in the n X n rook complement graph.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

Extensions

A347922 Number of minimal total dominating sets in the n X n rook complement graph.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions