A002465 Number of ways to place n nonattacking bishops on an n X n board. (Formerly M3616 N1467)

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

Vaclav Kotesovec, Table of n, a(n) for n = 0..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

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

Cf. A238258, A238260, A187235.

Main diagonal of A378590.

Sequence in context: A215242 A098620 A215266 * A248668 A079473 A145164

Adjacent sequences: A002462 A002463 A002464 * A002466 A002467 A002468

Keyword

nonn,nice

Author

N. J. A. Sloane

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