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 A002465 Number of ways to place n nonattacking bishops on an n X n board. (Formerly M3616 N1467) 23
 1, 4, 26, 260, 3368, 53744, 1022320, 22522960, 565532992, 15915225216, 496911749920, 17029582652416, 636101065346560, 25705530908501760, 1118038500044633088, 52054862490790200576, 2584158975023147147264 (list; graph; refs; listen; history; text; internal format)
 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 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. Sequence in context: A215242 A098620 A215266 * A248668 A079473 A145164 Adjacent sequences: A002462 A002463 A002464 * A002466 A002467 A002468 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

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Last modified February 1 07:16 EST 2023. Contains 359981 sequences. (Running on oeis4.)