A122749 Number of arrangements of n non-attacking bishops on an n X n board such that every square of the board is controlled by at least one bishop.

4, 2, 16, 44, 256, 768, 5184, 25344, 186624, 996480, 8294400, 57888000, 530841600, 4006195200, 40642560000, 367408742400, 4064256000000, 39358255104000, 474054819840000, 5254107586560000, 68263894056960000, 804207665479680000, 11242684107325440000

Offset: 2

Author

N. J. A. Sloane, Sep 25 2006

Keywords

• nonn

Links

• Vincenzo Librandi, Table of n, a(n) for n = 2..200
• R. W. Robinson, Counting arrangements of bishops, pp. 198-214 of Combinatorial Mathematics IV (Adelaide 1975), Lect. Notes Math., 560 (1976). [E_n, n >= 2.]

Crossrefs

Cf. A005635, A182333.

Programs

Maple

E:=proc(n) local k; if n mod 2 = 0 then k := n/2; if k mod 2 = 0 then RETURN( (k!*(k+2)/2)^2 ); else RETURN( ((k-1)!*(k+1)^2/2)^2 ); fi; else k := (n-1)/2; if k mod 2 = 0 then RETURN( ((k!)^2/12)*(3*k^3+16*k^2+18*k+8) ); else RETURN( ((k-1)!*(k+1)!/12)*(3*k^3+13*k^2-k-3) ); fi; fi; end;

Mathematica

Table[If[n==1,1,1/768*(2*(3*n^3+23*n^2+17*n+21)*(((n-1)/2)!)^2*(1-(-1)^n+2*Sin[(Pi*n)/2])-2*(3*n^3+17*n^2-47*n+3)*((n-3)/2)!*((n+1)/2)!*((-1)^n+2*Sin[(Pi*n)/2]-1)+3*(n+2)^4*((n/2-1)!)^2*((-1)^n-2*Cos[(Pi*n)/2]+1)+12*(n+4)^2*((n/2)!)^2*((-1)^n+2*Cos[(Pi*n)/2]+1))],{n,2,25}] (* Vaclav Kotesovec, Apr 26 2012 *)
a[n_] := Module[{k}, If[Mod[n, 2]==0, k = n/2; If[Mod[k, 2]==0, (k!*(k+2) /2)^2, ((k-1)!*(k+1)^2/2)^2], k = (n-1)/2; If[Mod[k, 2]==0, ((k!)^2/12)* (3*k^3+16*k^2+18*k+8), ((k-1)!*(k+1)!/12)*(3*k^3+13*k^2-k-3)]]];
Table[a[n], {n, 2, 25}] (* Jean-François Alcover, Jul 23 2022, after Maple code *)

Formula

From Andy Huchala, Mar 22 2024 (based on Mathematica code): (Start)

a(4*n) = (n+1)^2*((2*n)!)^2.

a(4*n+1) = (1/3)*(n+2)*(1+4*n+6*n^2)*((2*n)!)^2.

a(4*n+2) = 4*(n+1)^4*((2*n)!)^2.

a(4*n+3) = (1/3)*(3+17*n+22*n^2+6*n^3)*(2*n)!*(2*n+2)!. (End)

Extensions

New name from Vaclav Kotesovec, Apr 26 2012