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A005634 Bishops on an n X n board (see Robinson paper for details).

%I A005634 M3621 #26 Jan 20 2024 19:18:14
%S A005634 0,0,1,4,28,85,630,3096,23220,123952,1036080,7230828,66349440,
%T A005634 500721252,5080269600,45925520096,508031496000,4919774752448,
%U A005634 59256847036800,656763354386032,8532986691801600,100525956801641104,1405335512577427200,18431883446961030912
%N A005634 Bishops on an n X n board (see Robinson paper for details).
%C A005634 The problem of the bishops is to determine the number of inequivalent arrangements of n bishops on an n X n chessboard such that no bishop threatens another and every unoccupied square is threatened by some bishop. Two arrangements are considered equivalent if they are isomorphic by way of one of the eight symmetries of the chessboard. - _Jean-François Alcover_, Jul 24 2022 (after Robinson's paper).
%D A005634 R. W. Robinson, Counting arrangements of bishops, pp. 198-214 of Combinatorial Mathematics IV (Adelaide 1975), Lect. Notes Math., 560 (1976). [The sequence epsilon(n) page 212.]
%D A005634 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A005634 R. W. Robinson, <a href="/A000899/a000899_1.pdf">Counting arrangements of bishops</a>, pp. 198-214 of Combinatorial Mathematics IV (Adelaide 1975), Lect. Notes Math., 560 (1976). (Annotated scanned copy)
%p A005634 For Maple program see A005635.
%t A005634 e[n_] := Module[{k}, If[Mod[n, 2] == 0, k = n/2; If[Mod[k, 2] == 0, Return[(k!*(k + 2)/2)^2], Return[((k - 1)!*(k + 1)^2/2)^2]], k = (n - 1)/2; If[Mod[k, 2] == 0, Return[((k!)^2/12)*(3*k^3 + 16*k^2 + 18*k + 8)], Return[((k - 1)!*(k + 1)!/12)*(3*k^3 + 13*k^2 - k - 3)]]]];
%t A005634 c[n_] := Module[{k}, If[Mod[n, 2]==0, Return[0]]; k = (n-1)/2; If[Mod[k, 2] == 0, Return[k*2^(k-1)*((k/2)!)^2], Return[2^k*(((k+1)/2)!)^2]]];
%t A005634 d[n_] := d[n] = If[n <= 1, 1, d[n - 1] + (n - 1)*d[n - 2]];
%t A005634 B[n_] := B[n] = Which[n == 0 || n == -2, 1, OddQ[n], B[n - 1], True, 2*B[n - 2] + (n - 2)*B[n - 4]];
%t A005634 S[n_] := S[n] = Module[{k}, If[Mod[n, 2]==0, 0, k = (n-1)/2; B[k]*B[k+1]]];
%t A005634 M[n_] := Module[{k}, If[Mod[n, 2] == 0, k = n/2; If[Mod[k, 2] == 0, Return[k!*(k + 2)/2], Return[(k - 1)!*(k + 1)^2/2]], k = (n - 1)/2; Return[d[k]*d[k + 1]]]];
%t A005634 a[n_] := e[n]/8 - c[n]/8 + S[n]/4 - M[n]/4;
%t A005634 Table[a[n], {n, 2, 30}] (* _Jean-François Alcover_, Jul 23 2022, after Maple program in A005635 *)
%K A005634 nonn
%O A005634 2,4
%A A005634 _N. J. A. Sloane_
%E A005634 More terms from _N. J. A. Sloane_, Sep 28 2006