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Number of domino tilings (or dimer coverings) of the 2n X n grid.
1

%I #20 May 27 2022 15:34:34

%S 1,1,5,41,2245,185921,106912793,90124167441,540061286536921,

%T 4652799879944138561,289415868852204573601981,

%U 25545661075321867247577262777,16457725663617130715785831809325501,14905470663149838513993965664256435411841,99323759360556656337166635121447749135517599089

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

%H Alois P. Heinz, <a href="/A340535/b340535.txt">Table of n, a(n) for n = 0..63</a>

%F a(n) = A187596(2n,n) = A187596(n,2n) = A187616(2n,n).

%F a(n) = A099390(2n,n) = A099390(n,2n) for n >= 1.

%e a(2) = 5:

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

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

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

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%e .

%p b:= proc(m, n) option remember; local i, j, t, M;

%p M:= Matrix(n*m, shape=skewsymmetric);

%p for i to n do for j to m do t:= (i-1)*m+j;

%p if j<m then M[t, t+1]:= 1 fi;

%p if i<n then M[t, t+m]:= 1-2*irem(j, 2) fi

%p od od;

%p isqrt(LinearAlgebra[Determinant](M))

%p end:

%p a:= n-> b(2*n, n):

%p seq(a(n), n=0..15);

%t T[_?OddQ, _?OddQ] = 0;

%t 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 a[n_] := T[2n, n] // Round;

%t Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, May 27 2022 *)

%Y Cf. A004003, A099390, A187596, A187616.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Jan 10 2021