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

1, 1, 5, 41, 2245, 185921, 106912793, 90124167441, 540061286536921, 4652799879944138561, 289415868852204573601981, 25545661075321867247577262777, 16457725663617130715785831809325501, 14905470663149838513993965664256435411841, 99323759360556656337166635121447749135517599089

Offset

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..63

Formula

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

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

Example

a(2) = 5:
   .___.   .___.   .___.   .___.   .___.
   |___|   |___|   |___|   | | |   | | |
   |___|   |___|   | | |   |_|_|   |_|_|
   |___|   | | |   |_|_|   |___|   | | |
   |___|   |_|_|   |___|   |___|   |_|_|
.

Maple

b:= proc(m, n) option remember; local i, j, t, M;
       M:= Matrix(n*m, shape=skewsymmetric);
       for i to n do for j to m do t:= (i-1)*m+j;
          if j<m then M[t, t+1]:= 1 fi;
          if i<n then M[t, t+m]:= 1-2*irem(j, 2) fi
       od od;
       isqrt(LinearAlgebra[Determinant](M))
    end:
a:= n-> b(2*n, n):
seq(a(n), n=0..15);

Mathematica

T[_?OddQ, _?OddQ] = 0;
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}];
a[n_] := T[2n, n] // Round;
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, May 27 2022 *)

Crossrefs

Cf. A004003, A099390, A187596, A187616.

Sequence in context: A052113 A318294 A065035 * A145008 A216610 A025173

Adjacent sequences: A340532 A340533 A340534 * A340536 A340537 A340538

Keyword

nonn

Author

Alois P. Heinz, Jan 10 2021

Status

approved