A005635 - OEIS

A005635

Number of ways of placing n non-attacking bishops on an n X n board so that every square is attacked (or occupied).
(Formerly M2761)

1, 1, 1, 1, 3, 8, 36, 110, 666, 3250, 23436, 125198, 1037520, 7241272, 66360960, 500827928, 5080370400, 45926666984, 508032504000, 4919789029480, 59256857923200, 656763542278304, 8532986822438400, 100525959568386848, 1405335514253932800, 18431883489984091552

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OFFSET

0,5

COMMENTS

From Vaclav Kotesovec, Apr 26 2012: (Start)

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

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.

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.

(End)

REFERENCES

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

LINKS

N. J. A. Sloane, Table of n, a(n) for n = 0..250

Jean-François Alcover, Mathematica program.

R. W. Robinson, Counting arrangements of bishops, pp. 198-214 of Combinatorial Mathematics IV (Adelaide 1975), Lect. Notes Math., 560 (1976).

R. W. Robinson, Counting arrangements of bishops, pp. 198-214 of Combinatorial Mathematics IV (Adelaide 1975), Lect. Notes Math., 560 (1976). (Annotated scanned copy)

MAPLE

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

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

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

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

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

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

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

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

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

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

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

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

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

CROSSREFS

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

Sequence in context: A294385 A020099 A182392 * A026649 A148919 A087905

Adjacent sequences: A005632 A005633 A005634 * A005636 A005637 A005638