A201540 Number of ways to place n nonattacking knights on an n X n board.

1, 6, 36, 412, 9386, 257318, 8891854, 379978716, 19206532478, 1120204619108, 74113608972922, 5483225594409823, 448414229054798028, 40154319792412218900, 3906519894750904583838

Offset: 1

Author

Vaclav Kotesovec, Dec 02 2011

Keywords

• nonn
• nice
• hard
• more

Comments

a(n) = A244081(n,n). - Alois P. Heinz, Jun 19 2014

Links

• V. Kotesovec, Non-attacking chess pieces, 6ed, 2013, p. 293.

Crossrefs

Cf. A172132, A172134, A172135, A172136, A178499, A244081.

Cf. A244284, A201511, A201861, A201513, A141243.

Programs

Mathematica

b[n_, l_] := b[n, l] = Module[{d, f, g, k}, d = Length[l]/3; f = False; Which[n == 0, 1, l[[1 ;; d]] == Array[f&, d], b[n - 1, Join[l[[d + 1 ;; 3*d]], Array[True&, d]]], True, For[k = 1, ! l[[k]], k++]; g = ReplacePart[l, k -> f];
     If[k > 1, g = ReplacePart[g, 2*d - 1 + k -> f]];
     If[k < d, g = ReplacePart[g, 2*d + 1 + k -> f]];
     If[k > 2, g = ReplacePart[g, d - 2 + k -> f]];
     If[k < d - 1, g = ReplacePart[g, d + 2 + k -> f]];
     Expand[b[n, ReplacePart[l, k -> f]] + b[n, g]*x]]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][ b[n, Array[True&, n*3]]];
a[n_] := T[n][[n + 1]];
Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 1, 10}] (* Jean-François Alcover, Mar 29 2016, after Alois P. Heinz's code for A244081 *)

Formula

a(n) ~ n^(2n)/n!*exp(-9/2). - Vaclav Kotesovec, Nov 29 2011

Extensions

a(11) from Alois P. Heinz, Jun 19 2014

a(12)-a(13) from Vaclav Kotesovec, Jun 21 2014

a(14) from Vaclav Kotesovec, Aug 26 2016

a(15) from Vaclav Kotesovec, May 26 2021