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A122749
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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.
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7
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4, 2, 16, 44, 256, 768, 5184, 25344, 186624, 996480, 8294400, 57888000, 530841600, 4006195200, 40642560000, 367408742400, 4064256000000, 39358255104000, 474054819840000, 5254107586560000, 68263894056960000, 804207665479680000, 11242684107325440000
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OFFSET
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2,1
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LINKS
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FORMULA
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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)
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MAPLE
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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;
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MATHEMATICA
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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)]]];
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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