A189002 Number of domino tilings of the n X n grid with upper left corner removed iff n is odd.

1, 1, 2, 4, 36, 192, 6728, 100352, 12988816, 557568000, 258584046368, 32565539635200, 53060477521960000, 19872369301840986112, 112202208776036178000000, 126231322912498539682594816, 2444888770250892795802079170816, 8326627661691818545121844900397056

Offset

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..50

Index entries for sequences related to dominoes

Formula

a(n) = A189006(n,n).

Example

a(3) = 4 because there are 4 domino tilings of the 3 X 3 grid with upper left corner removed:
  . .___.  . .___.  . .___.  . .___.
  ._|___|  ._|___|  ._| | |  ._|___|
  | |___|  | | | |  | |_|_|  |___| |
  |_|___|  |_|_|_|  |_|___|  |___|_|

Mathematica

A[1, 1] = 1;
A[m_, n_] := A[m, n] = Module[{i, j, s, t, M}, Which[m == 0 || n == 0, 1, m < n, A[n, m], True, s = Mod[n*m, 2]; M[i_, j_] /; j < i := -M[j, i]; M[_, _] = 0; For[i = 1, i <= n, i++, For[j = 1, j <= m, j++, t = (i - 1)*m + j - s; If[i > 1 || j > 1 || s == 0, If[j < m, M[t, t + 1] = 1]; If[i < n, M[t, t + m] = 1 - 2*Mod[j, 2]]]]]; Sqrt[Det[Array[M, {n*m - s, n*m - s}]] ]]];
a[n_] := A[n, n];
a /@ Range[0, 17] (* Jean-François Alcover, Feb 27 2020, after Alois P. Heinz in A189006 *)

Crossrefs

Main diagonal of A189006.

Bisection gives: A004003 (even part), A007341 (odd part).

Sequence in context: A277091 A199495 A182965 * A304558 A215251 A052716

Adjacent sequences: A188999 A189000 A189001 * A189003 A189004 A189005

Keyword

nonn

Author

Alois P. Heinz, Apr 15 2011

Status

approved