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A239273 Number of domicule tilings of a 2n X 2n square grid. 4
1, 3, 280, 3037561, 3263262629905, 326207195516663381931, 3011882198082438957330143630563, 2565014347691062208319404612723752103028288, 201442620359313683494245316355883565275531844406384955392, 1458834332808489549111708247664894524221330758005874053074138540424018259 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
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
LINKS
Eric Weisstein's World of Mathematics, King Graph
Wikipedia, King's graph
FORMULA
a(n) = A239264(2n,2n).
EXAMPLE
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|
+-------+ +-------+ +-------+ +-------+ +-------+ ...
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 *)
CROSSREFS
Even bisection of main diagonal of A239264.
Sequence in context: A263884 A096126 A057599 * A054583 A139984 A057631
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Mar 13 2014
EXTENSIONS
a(8) from Alois P. Heinz, Sep 30 2014
a(9) from Alois P. Heinz, Nov 23 2018
STATUS
approved

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Last modified July 15 12:56 EDT 2024. Contains 374332 sequences. (Running on oeis4.)