%I M2761 #31 Jul 23 2022 19:26:01

%S 1,1,1,1,3,8,36,110,666,3250,23436,125198,1037520,7241272,66360960,

%T 500827928,5080370400,45926666984,508032504000,4919789029480,

%U 59256857923200,656763542278304,8532986822438400,100525959568386848,1405335514253932800,18431883489984091552

%N Number of ways of placing n non-attacking bishops on an n X n board so that every square is attacked (or occupied).

%C From _Vaclav Kotesovec_, Apr 26 2012: (Start)

%C This sequence gives (according to the article by Robinson) the number of inequivalent solutions.

%C For the total number of all arrangements of n non-attacking bishops such that every square of the board is controlled by at least one bishop, see A122749.

%C For the total number of all arrangements of n bishops (in any position) such that every square of the board is controlled by at least one bishop, see A182333.

%C (End)

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H N. J. A. Sloane, <a href="/A005635/b005635.txt">Table of n, a(n) for n = 0..250</a>

%H Jean-François Alcover, <a href="/A005635/a005635.txt">Mathematica program</a>.

%H R. W. Robinson, <a href="http://dx.doi.org/10.1007/BFb0097382">Counting arrangements of bishops</a>, pp. 198-214 of Combinatorial Mathematics IV (Adelaide 1975), Lect. Notes Math., 560 (1976).

%H 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 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; # Gives A122749

%p unprotect(D); D:=proc(n) option remember; if n <= 1 then 1 else D(n-1)+(n-1)*D(n-2); fi; end; # Gives A000085

%p C:=proc(n) local k; if n mod 2 = 0 then RETURN(0); fi; k:=(n-1)/2; if k mod 2 = 0 then RETURN( k*2^(k-1)*((k/2)!)^2 ); else RETURN( 2^k*(((k+1)/2)!)^2 ); fi; end; # Gives A122693

%p Q:=proc(n) local m; if n mod 8 <> 1 then RETURN(0); fi; m:=(n-1)/8; ((2*m)!)^2/(m!)^2; end; # Gives A122747

%p M:=proc(n) local k; if n mod 2 = 0 then k:=n/2; if k mod 2 = 0 then RETURN( k!*(k+2)/2 ); else RETURN( (k-1)!*(k+1)^2/2 ); fi; else k:=(n-1)/2; RETURN(D(k)*D(k+1)); fi; end; # Gives A122748

%p a:=n-> if n <= 1 then RETURN(1) else E(n)/8 + C(n)/8 + Q(n)/4 + M(n)/4; fi; # Gives A005635

%p # The following additional Maple programs produce A123071, A005631, A123072, A005633, A005632, A005634

%p S:=proc(n) local k; if n mod 2 = 0 then RETURN(0) else k:=(n-1)/2; RETURN(B(k)*B(k+1)); fi; end; # Gives A123071

%p psi:=n->S(n)/2; # Gives A005631

%p zeta:=n->Q(n)/2; # Gives A123072

%p mu:=n->(M(n)-S(n))/2; # Gives A005633

%p chi:=n->(C(n)-S(n)-Q(n))/4; # Gives A005632

%p eps:=n->E(n)/8-C(n)/8+S(n)/4-M(n)/4; # Gives A005634

%Y Cf. A005631, A005632, A005633, A005634, A122749, A123071, A123072, A182333.

%K easy,nonn

%O 0,5

%A _N. J. A. Sloane_

%E Entry revised by _N. J. A. Sloane_, Sep 25 2006