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

1,3

Links

Table of n, a(n) for n=1..23.

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

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.

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 *)

Prog

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

Crossrefs

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

Sequence in context: A150935 A150936 A109719 * A334443 A301868 A288958

Adjacent sequences: A263682 A263683 A263684 * A263686 A263687 A263688

Keyword

nonn,nice

Author

Max Alekseyev, Oct 31 2015

Status

approved