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

2,1

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.]

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)

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 *)

Crossrefs

Cf. A005635, A182333.

Sequence in context: A182872 A137393 A356569 * A189741 A303142 A328695

Adjacent sequences: A122746 A122747 A122748 * A122750 A122751 A122752

Keyword

nonn

Author

N. J. A. Sloane, Sep 25 2006

Extensions

New name from Vaclav Kotesovec, Apr 26 2012

Status

approved