A263685 Number of inequivalent placements of n nonattacking rooks on n X n board up to rotations of the board.

1, 1, 2, 9, 33, 192, 1272, 10182, 90822, 908160, 9980160, 119761980, 1556766780, 21794734080, 326918753280, 5230700053320, 88921859605320, 1600593472880640, 30411275148656640, 608225502973147920, 12772735543856347920, 281000181964839321600, 6463004184741681561600

Offset: 1

Author

Max Alekseyev, Oct 31 2015

Keywords

• nonn
• nice

Links

• R. W. Robinson, Counting arrangements of bishops, Lect. Notes Math. 560 (1976), 198-214.

Crossrefs

Cf. A000085, A000407, A000899, A000900, A000901, A000902, A000903.

Programs

Mathematica

a[n_] := (r=Mod[n, 4]; m=(n-r)/4; q=Quotient[n, 2]; n! + q!*2^q + 2*If[r <= 1, (2m)!/m!, 0])/4; Array[a, 23] (* Jean-François Alcover, Dec 06 2015, adapted from PARI *)

PARI

{ a(n) = ( n! + (n\2)! * 2^(n\2) + 2*if(n%4<=1, (2*(n\4))!/(n\4)! ) )/4; }

Formula

For n=4m or n=4m+1, a(n) = (n! + (2m)!*2^(2*m) + (2m)!/m!)/4.

For n=4m+2 or n=4m+3, a(n) = (n! + (2m+1)!*2^(2*m+1))/4.

a(n) = 2*A000903(n) - A000900(n) - A000902(floor(n/2)).

For n>1, a(n) = 2*A000903(n) - A000085(n)/2.

a(n) = (P(n)+G(n)+2*R(n))/4, where P,G,R are defined in Robinson (1976). See also Maple code in A000903.