OFFSET
1,2
COMMENTS
The old name of this sequence was wrong. It was corrected by Vaclav Kotesovec, Feb 19 2011. Kotesovec remarks that the maximal number of nonattacking bishops on an n X n board is 2n-2, and there are 2^n ways to place them. See the Kotesovec link.
REFERENCES
W. Ahrens, Mathematische Unterhaltungen und Spiele. Teubner, Leipzig, Vol. 1, 3rd ed., 1921; Vol. 2, 2nd ed., 1918. See Vol. 1, p. 271.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
N. Vilenkin, Populyarnaja kombinatorika, 1972, p. 166.
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 1..375
W. Ahrens, Mathematische Unterhaltungen und Spiele, Leipzig: B. G. Teubner, 1901.
S. E. Arshon, Solution of one combinatorial problem [in Russian], Matematicheskoe prosveshchenie, Ser. 1, 8, 1936, pp. 24-29.
D. Atkinson, Solution to the n-Bishops problem of trying to place n identical bishops on an n x n chessboard. [Broken link?]
Vaclav Kotesovec, Non-attacking chess pieces, 6ed, 2013, pp. 242-252.
J. Perott, Sur le problème des fous, Bulletin de la société mathématique de France, Tome XI, 1883, p. 173-186.
Eric Weisstein's World of Mathematics, Bishops Problem.
FORMULA
Asymptotic: a(n)/(n-1)! ~ 0.631266 * 3.08827^n. - Vaclav Kotesovec, Mar 23 2011
The second constant is 2/(z*(2-z)) = 3.0882773047417401791158400820254..., where z is the root z=1.593624260040... of the equation exp(z)*(2-z)=2. - Vaclav Kotesovec, May 27 2011
EXAMPLE
a(3) = 26: ways to place 3 nonattacking bishops on a 3 X 3 board:
XXX XXO XXO XOX OXO
OOO OOO OOO OOO OXO
OOO XOO OXO OXO OXO
(4) (8) (8) (4) (2)
MATHEMATICA
peven[i_]:=(Sum[(-1)^j*Binomial[n-i-1, j]/(n-i-1)!*(n-i+1-j)^(n/2)*(n-i-j)^(n/2-1), {j, 0, n-i-1}]);
poddblack[i_]:=(Sum[(-1)^j*Binomial[n-i-1, j]/(n-i-1)!*(n-i+1-j)^((n+1)/2)*(n-i-j)^((n-3)/2), {j, 0, n-i-1}]);
poddwhite[i_]:=(Sum[(-1)^j*Binomial[n-i-1, j]/(n-i-1)!*(n-i+1-j)^((n-1)/2)*(n-i-j)^((n-1)/2), {j, 0, n-i-1}]);
Table[If[n==1, 1, Sum[If[EvenQ[n], peven[i]*peven[n-i], poddblack[i]*poddwhite[n-i]], {i, 1, n-1}]], {n, 1, 50}]
(* Alternative formula with Stirling numbers of the second kind: *)
Table[If[n==1, 1, Sum[Sum[Binomial[Floor[(n+1)/2], j] * StirlingS2[j+Floor[n/2], n-i], {j, 0, Floor[(n+1)/2]}] * Sum[Binomial[Floor[n/2], j] * StirlingS2[j+Floor[(n+1)/2], i], {j, 0, Floor[n/2]}], {i, 1, n-1}]], {n, 1, 50}] (* Vaclav Kotesovec, Mar 23 2011 *)
CROSSREFS
KEYWORD
nonn,nice
AUTHOR
EXTENSIONS
More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Nov 20 2006
Definition corrected by Vaclav Kotesovec, Feb 19 2011
Terms a(11)-a(17) from Vaclav Kotesovec, Mar 09 2011
STATUS
approved