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A005635
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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)
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8
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,5
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REFERENCES
| R. W. Robinson, Counting arrangements of bishops, pp. 198-214 of Combinatorial Mathematics IV (Adelaide 1975), Lect. Notes Math., 560 (1976).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| N. J. A. Sloane, Table of n, a(n) for n = 0..250
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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
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CROSSREFS
| Cf. A005631, A005632, A005633, A005634, A123071, A123072.
Sequence in context: A148918 A194090 A020099 * A026649 A148919 A087905
Adjacent sequences: A005632 A005633 A005634 * A005636 A005637 A005638
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KEYWORD
| easy,nonn
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| Entry revised by N. J. A. Sloane (njas(AT)research.att.com), Sep 25 2006
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