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A002465
Number of ways to place n nonattacking bishops on an n X n board.
24
1, 1, 4, 26, 260, 3368, 53744, 1022320, 22522960, 565532992, 15915225216, 496911749920, 17029582652416, 636101065346560, 25705530908501760, 1118038500044633088, 52054862490790200576, 2584158975023147147264
OFFSET
0,3
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
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.
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
For constants see A238258 and A238260. - Vaclav Kotesovec, Feb 21 2014
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
Main diagonal of A378590.
Sequence in context: A215242 A098620 A215266 * A248668 A079473 A145164
KEYWORD
nonn,nice
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
a(0)=1 prepended by Alois P. Heinz, Dec 01 2024
STATUS
approved