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A239273 Number of domicule tilings of a 2n X 2n square grid.

%I A239273 #29 Feb 16 2025 08:33:21
%S A239273 1,3,280,3037561,3263262629905,326207195516663381931,
%T A239273 3011882198082438957330143630563,
%U A239273 2565014347691062208319404612723752103028288,201442620359313683494245316355883565275531844406384955392,1458834332808489549111708247664894524221330758005874053074138540424018259
%N A239273 Number of domicule tilings of a 2n X 2n square grid.
%C A239273 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 A239273 Number of perfect matchings in the 2n X 2n kings graph. - _Andrew Howroyd_, Apr 07 2016
%H A239273 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/KingGraph.html">King Graph</a>
%H A239273 Wikipedia, <a href="https://en.wikipedia.org/wiki/King%27s_graph">King's graph</a>
%F A239273 a(n) = A239264(2n,2n).
%e A239273 a(1) = 3:
%e A239273   +---+   +---+   +---+
%e A239273   |o o|   |o o|   |o-o|
%e A239273   || ||   | X |   |   |
%e A239273   |o o|   |o o|   |o-o|
%e A239273   +---+   +---+   +---+.
%e A239273 a(2) = 280:
%e A239273   +-------+ +-------+ +-------+ +-------+ +-------+
%e A239273   |o o o-o| |o o o-o| |o-o o-o| |o o o o| |o o-o o|
%e A239273   | X     | | X     | |       | | X  | || | \   / |
%e A239273   |o o o o| |o o o o| |o o o o| |o o o o| |o o o o|
%e A239273   |   /  || |   / / | ||  X  || |       | ||     ||
%e A239273   |o o o o| |o o o o| |o o o o| |o-o o o| |o o o o|
%e A239273   ||    \ | ||     || |       | |     X | | / /   |
%e A239273   |o o-o o| |o o-o o| |o-o o-o| |o-o o o| |o o o-o|
%e A239273   +-------+ +-------+ +-------+ +-------+ +-------+ ...
%t A239273 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 A239273 A[n_, k_] := If[Mod[n*k, 2]>0, 0, If[k>n, A[k, n], b[n, Array[True&, k*2]]]];
%t A239273 a[n_] := A[2n, 2n];
%t A239273 Table[Print[n]; a[n], {n, 0, 7}] (* _Jean-François Alcover_, Sep 16 2019, after _Alois P. Heinz_ in A239264 *)
%Y A239273 Even bisection of main diagonal of A239264.
%Y A239273 Cf. A004003, A243510, A243424, A220638.
%K A239273 nonn
%O A239273 0,2
%A A239273 _Alois P. Heinz_, Mar 13 2014
%E A239273 a(8) from _Alois P. Heinz_, Sep 30 2014
%E A239273 a(9) from _Alois P. Heinz_, Nov 23 2018