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

1, 3, 280, 3037561, 3263262629905, 326207195516663381931, 3011882198082438957330143630563, 2565014347691062208319404612723752103028288, 201442620359313683494245316355883565275531844406384955392, 1458834332808489549111708247664894524221330758005874053074138540424018259

Offset: 0

Author

Alois P. Heinz, Mar 13 2014

Keywords

• nonn

Comments

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.

Number of perfect matchings in the 2n X 2n kings graph. - Andrew Howroyd, Apr 07 2016

Examples

			a(1) = 3:
  +---+   +---+   +---+
  |o o|   |o o|   |o-o|
  || ||   | X |   |   |
  |o o|   |o o|   |o-o|
  +---+   +---+   +---+.
a(2) = 280:
  +-------+ +-------+ +-------+ +-------+ +-------+
  |o o o-o| |o o o-o| |o-o o-o| |o o o o| |o o-o o|
  | X     | | X     | |       | | X  | || | \   / |
  |o o o o| |o o o o| |o o o o| |o o o o| |o o o o|
  |   /  || |   / / | ||  X  || |       | ||     ||
  |o o o o| |o o o o| |o o o o| |o-o o o| |o o o o|
  ||    \ | ||     || |       | |     X | | / /   |
  |o o-o o| |o o-o o| |o-o o-o| |o-o o o| |o o o-o|
  +-------+ +-------+ +-------+ +-------+ +-------+ ...
		

Links

• Eric Weisstein's World of Mathematics, King Graph
• Wikipedia, King's graph

Crossrefs

Even bisection of main diagonal of A239264.

Cf. A004003, A243510, A243424, A220638.

Programs

Mathematica

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]]];
A[n_, k_] := If[Mod[n*k, 2]>0, 0, If[k>n, A[k, n], b[n, Array[True&, k*2]]]];
a[n_] := A[2n, 2n];
Table[Print[n]; a[n], {n, 0, 7}] (* Jean-François Alcover, Sep 16 2019, after Alois P. Heinz in A239264 *)

Formula

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

Extensions

a(8) from Alois P. Heinz, Sep 30 2014

a(9) from Alois P. Heinz, Nov 23 2018