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A189004 Number of domino tilings of the 7 X n grid with upper left corner removed iff n is odd.

%I A189004 #15 Mar 03 2024 10:07:51
%S A189004 1,1,21,56,781,2415,31529,100352,1292697,4140081,53175517,170537640,
%T A189004 2188978117,7022359583,90124167441,289143013376,3710708201969,
%U A189004 11905151192865,152783289861989,490179860527896,6290652543875133
%N A189004 Number of domino tilings of the 7 X n grid with upper left corner removed iff n is odd.
%H A189004 Alois P. Heinz, <a href="/A189004/b189004.txt">Table of n, a(n) for n = 0..400</a>
%H A189004 <a href="/index/Do#domino">Index entries for sequences related to dominoes</a>
%H A189004 <a href="/index/Rec#order_16">Index entries for linear recurrences with constant coefficients</a>, signature (0, 56, 0, -672, 0, 2632, 0, -4094, 0, 2632, 0, -672, 0, 56, 0, -1).
%F A189004 G.f.: -(x^14-x^13-35*x^12+277*x^10 +49*x^9-727*x^8 -112*x^7+727*x^6 +49*x^5-277*x^4 +35*x^2-x-1) / (x^16-56*x^14 +672*x^12-2632*x^10 +4094*x^8-2632*x^6 +672*x^4-56*x^2+1).
%t A189004 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 A189004 a[n_] := A[7, n];
%t A189004 a /@ Range[0, 20] (* _Jean-François Alcover_, Feb 27 2020, after _Alois P. Heinz_ in A189006 *)
%Y A189004 7th row of array A189006.
%Y A189004 Bisection gives: A028469 (even part), A003696 (odd part).
%K A189004 nonn,easy
%O A189004 0,3
%A A189004 _Alois P. Heinz_, Apr 15 2011