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A005635 Number of ways of placing n non-attacking bishops on an n X n board so that every square is attacked (or occupied).

%I A005635 M2761 #31 Jul 23 2022 19:26:01
%S A005635 1,1,1,1,3,8,36,110,666,3250,23436,125198,1037520,7241272,66360960,
%T A005635 500827928,5080370400,45926666984,508032504000,4919789029480,
%U A005635 59256857923200,656763542278304,8532986822438400,100525959568386848,1405335514253932800,18431883489984091552
%N A005635 Number of ways of placing n non-attacking bishops on an n X n board so that every square is attacked (or occupied).
%C A005635 From _Vaclav Kotesovec_, Apr 26 2012: (Start)
%C A005635 This sequence gives (according to the article by Robinson) the number of inequivalent solutions.
%C A005635 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 A005635 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 A005635 (End)
%D A005635 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A005635 N. J. A. Sloane, <a href="/A005635/b005635.txt">Table of n, a(n) for n = 0..250</a>
%H A005635 Jean-François Alcover, <a href="/A005635/a005635.txt">Mathematica program</a>.
%H A005635 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 A005635 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 A005635 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 A005635 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 A005635 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 A005635 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 A005635 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 A005635 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 A005635 # The following additional Maple programs produce A123071, A005631, A123072, A005633, A005632, A005634
%p A005635 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 A005635 psi:=n->S(n)/2; # Gives A005631
%p A005635 zeta:=n->Q(n)/2; # Gives A123072
%p A005635 mu:=n->(M(n)-S(n))/2; # Gives A005633
%p A005635 chi:=n->(C(n)-S(n)-Q(n))/4; # Gives A005632
%p A005635 eps:=n->E(n)/8-C(n)/8+S(n)/4-M(n)/4; # Gives A005634
%Y A005635 Cf. A005631, A005632, A005633, A005634, A122749, A123071, A123072, A182333.
%K A005635 easy,nonn
%O A005635 0,5
%A A005635 _N. J. A. Sloane_
%E A005635 Entry revised by _N. J. A. Sloane_, Sep 25 2006