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Number of domicule tilings of a 2n X 2n square grid.
4

%I #28 Sep 16 2019 08:23:24

%S 1,3,280,3037561,3263262629905,326207195516663381931,

%T 3011882198082438957330143630563,

%U 2565014347691062208319404612723752103028288,201442620359313683494245316355883565275531844406384955392,1458834332808489549111708247664894524221330758005874053074138540424018259

%N Number of domicule tilings of a 2n X 2n square grid.

%C A domicule is either a domino or it is formed by the union of two neighboring unit squares connected via their corners. In a tiling the connections of two domicules are allowed to cross each other.

%C Number of perfect matchings in the 2n X 2n kings graph. - _Andrew Howroyd_, Apr 07 2016

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/KingGraph.html">King Graph</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/King%27s_graph">King's graph</a>

%F a(n) = A239264(2n,2n).

%e a(1) = 3:

%e +---+ +---+ +---+

%e |o o| |o o| |o-o|

%e || || | X | | |

%e |o o| |o o| |o-o|

%e +---+ +---+ +---+.

%e a(2) = 280:

%e +-------+ +-------+ +-------+ +-------+ +-------+

%e |o o o-o| |o o o-o| |o-o o-o| |o o o o| |o o-o o|

%e | X | | X | | | | X | || | \ / |

%e |o o o o| |o o o o| |o o o o| |o o o o| |o o o o|

%e | / || | / / | || X || | | || ||

%e |o o o o| |o o o o| |o o o o| |o-o o o| |o o o o|

%e || \ | || || | | | X | | / / |

%e |o o-o o| |o o-o o| |o-o o-o| |o-o o o| |o o o-o|

%e +-------+ +-------+ +-------+ +-------+ +-------+ ...

%t b[n_, l_List] := b[n, l] = Module[{d = Length[l]/2, f = False, k}, Which[n == 0, 1, l[[1 ;; d]] == Array[f &, d], b[n - 1, Join[l[[d + 1 ;; 2*d]], Array[True &, d]]], True, For[k = 1, ! l[[k]], k++]; If[k < d && n > 1 && l[[k + d + 1]], b[n, ReplacePart[l, {k -> f, k + d + 1 -> f}]], 0] + If[k > 1 && n > 1 && l[[k + d - 1]], b[n, ReplacePart[l, {k -> f, k + d - 1 -> f}]], 0] + If[n > 1 && l[[k + d]], b[n, ReplacePart[l, {k -> f, k + d -> f}]], 0] + If[k < d && l[[k + 1]], b[n, ReplacePart[l, {k -> f, k + 1 -> f}]], 0]]];

%t A[n_, k_] := If[Mod[n*k, 2]>0, 0, If[k>n, A[k, n], b[n, Array[True&, k*2]]]];

%t a[n_] := A[2n, 2n];

%t Table[Print[n]; a[n], {n, 0, 7}] (* _Jean-François Alcover_, Sep 16 2019, after _Alois P. Heinz_ in A239264 *)

%Y Even bisection of main diagonal of A239264.

%Y Cf. A004003, A243510, A243424, A220638.

%K nonn

%O 0,2

%A _Alois P. Heinz_, Mar 13 2014

%E a(8) from _Alois P. Heinz_, Sep 30 2014

%E a(9) from _Alois P. Heinz_, Nov 23 2018