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A340535 Number of domino tilings (or dimer coverings) of the 2n X n grid.

%I A340535 #20 May 27 2022 15:34:34
%S A340535 1,1,5,41,2245,185921,106912793,90124167441,540061286536921,
%T A340535 4652799879944138561,289415868852204573601981,
%U A340535 25545661075321867247577262777,16457725663617130715785831809325501,14905470663149838513993965664256435411841,99323759360556656337166635121447749135517599089
%N A340535 Number of domino tilings (or dimer coverings) of the 2n X n grid.
%H A340535 Alois P. Heinz, <a href="/A340535/b340535.txt">Table of n, a(n) for n = 0..63</a>
%F A340535 a(n) = A187596(2n,n) = A187596(n,2n) = A187616(2n,n).
%F A340535 a(n) = A099390(2n,n) = A099390(n,2n) for n >= 1.
%e A340535 a(2) = 5:
%e A340535    .___.   .___.   .___.   .___.   .___.
%e A340535    |___|   |___|   |___|   | | |   | | |
%e A340535    |___|   |___|   | | |   |_|_|   |_|_|
%e A340535    |___|   | | |   |_|_|   |___|   | | |
%e A340535    |___|   |_|_|   |___|   |___|   |_|_|
%e A340535 .
%p A340535 b:= proc(m, n) option remember; local i, j, t, M;
%p A340535        M:= Matrix(n*m, shape=skewsymmetric);
%p A340535        for i to n do for j to m do t:= (i-1)*m+j;
%p A340535           if j<m then M[t, t+1]:= 1 fi;
%p A340535           if i<n then M[t, t+m]:= 1-2*irem(j, 2) fi
%p A340535        od od;
%p A340535        isqrt(LinearAlgebra[Determinant](M))
%p A340535     end:
%p A340535 a:= n-> b(2*n, n):
%p A340535 seq(a(n), n=0..15);
%t A340535 T[_?OddQ, _?OddQ] = 0;
%t A340535 T[m_, n_] := Product[2(2+Cos[2 j Pi/(m+1)]+Cos[2 k Pi/(n+1)]), {k, 1, n/2}, {j, 1, m/2}];
%t A340535 a[n_] := T[2n, n] // Round;
%t A340535 Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, May 27 2022 *)
%Y A340535 Cf. A004003, A099390, A187596, A187616.
%K A340535 nonn
%O A340535 0,3
%A A340535 _Alois P. Heinz_, Jan 10 2021