A189002 - OEIS

%I #21 Jul 14 2020 15:26:35

%S 1,1,2,4,36,192,6728,100352,12988816,557568000,258584046368,

%T 32565539635200,53060477521960000,19872369301840986112,

%U 112202208776036178000000,126231322912498539682594816,2444888770250892795802079170816,8326627661691818545121844900397056

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

%H Alois P. Heinz, <a href="/A189002/b189002.txt">Table of n, a(n) for n = 0..50</a>

%H <a href="/index/Do#domino">Index entries for sequences related to dominoes</a>

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

%e a(3) = 4 because there are 4 domino tilings of the 3 X 3 grid with upper left corner removed:

%e   . .___.  . .___.  . .___.  . .___.

%e   ._|___|  ._|___|  ._| | |  ._|___|

%e   | |___|  | | | |  | |_|_|  |___| |

%e   |_|___|  |_|_|_|  |_|___|  |___|_|

%t A[1, 1] = 1;

%t 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}]] ]]];

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

%t a /@ Range[0, 17] (* _Jean-François Alcover_, Feb 27 2020, after _Alois P. Heinz_ in A189006 *)

%Y Main diagonal of A189006.

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

%K nonn

%O 0,3

%A _Alois P. Heinz_, Apr 15 2011