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%I #45 Aug 24 2020 11:56:35
%S 1,2,16,94,1365,55213,3368146,394631712,101693175442,50929053498909,
%T 48988729226134301,96325314726538906164,375615195988659173454092,
%U 2933480442104347575000834468,45480806737377995771543610802659,1422902021111889804120495149240353936
%N Number of ways to place non-attacking knights on the n X n board.
%C The maximum number of non-attacking knights is given by A030978.
%C Also the number of vertex covers and independent vertex sets in the n X n knight graph.
%H Vaclav Kotesovec, <a href="https://oeis.org/wiki/User:Vaclav_Kotesovec">Non-attacking chess pieces</a>, 6ed, 2013, p. 69-72, 283-285.
%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/IndependentVertexSet.html">Independent Vertex Set</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/KnightGraph.html">Knight Graph</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/VertexCover.html">Vertex Cover</a>
%t 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];
%t If[k > 1, g = ReplacePart[g, 2*d - 1 + k -> f]];
%t If[k < d, g = ReplacePart[g, 2*d + 1 + k -> f]];
%t If[k > 2, g = ReplacePart[g, d - 2 + k -> f]];
%t If[k < d - 1, g = ReplacePart[g, d + 2 + k -> f]];
%t Expand[b[n, ReplacePart[l, k -> f]] + b[n, g]*x]]];
%t a[n_] := Function[p, Sum[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][ b[n, Array[True &, n*3]]];
%t Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 0, 10}] (* _Jean-François Alcover_, Mar 29 2016, after _Alois P. Heinz_'s code for A244081 *)
%Y Cf. A182407, A182408, A201540.
%Y Row sums of A244081.
%K hard,nonn,nice
%O 0,2
%A _Max Alekseyev_, Jun 17 2008
%E a(8)-a(13) from _R. H. Hardin_, Aug 25 2008
%E a(0) from _Alois P. Heinz_, Jun 19 2014
%E a(14) from _Hiroaki Yamanouchi_, Aug 28 2014
%E a(15) from _Hiroaki Yamanouchi_, Aug 29 2014
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