A340535
Number of domino tilings (or dimer coverings) of the 2n X n grid.
Original entry on oeis.org
1, 1, 5, 41, 2245, 185921, 106912793, 90124167441, 540061286536921, 4652799879944138561, 289415868852204573601981, 25545661075321867247577262777, 16457725663617130715785831809325501, 14905470663149838513993965664256435411841, 99323759360556656337166635121447749135517599089
Offset: 0
a(2) = 5:
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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 b(2*n, n):
seq(a(n), n=0..15);
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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 *)
A347054
Number of domino tilings of a 32 X n rectangle.
Original entry on oeis.org
1, 1, 3524578, 1117014753, 170220478472105, 224916047725262248, 12348080425980866090537, 30648981125778378496845537, 1010618564986361239515088848178, 3596059736380751648485086101179655, 87171995375835553001398855677616476448, 391978133958466896956216157693001644153072
Offset: 0
- A. M. Magomedov, T. A. Magomedov, S. A. Lawrencenko, Mutually-recursive formulas for enumerating partitions of the rectangle, [in Russian, English summary], Prikl. Diskretn. Mat., 46 (2019), 108-121. DOI: 10.17223/20710410/46/9
- A. M. Magomedov and S. A. Lavrenchenko, Computational aspects of the partition enumeration problem, [in Russian, English summary], Dagestan Electronic Mathematical Reports, 14 (2020), 1-21. DOI: 10.31029/demr.14.1
- M. E. Fisher, Statistical mechanics of dimers on a plane lattice, Phys. Rev., 124 (1961), 1664-1672.
- P. W. Kasteleyn, The statistics of dimers on a lattice: I. The number of dimer arrangements on a quadratic lattice, Physica, 27 (1961), 1209-1225.
- P. W. Kasteleyn, Dimer statistics and phase transitions, J. Math. Phys., 4 (1963), 287-293.
- V.-H. Nguyen, K. Perrot, M. Vallet, NP-completeness of the game Kingdomino{TM}, Theoret. Comput. Sci., 822 (2020), 23-35.
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Do[ P=1;
Do[P=P*4*(Cos[Pi*i/(n+1)]^2+Cos[Pi*j/33]^2), {i,1,n/2}, {j,1,16}];
Print["n=", n ,":", Round[P]], {n,1,11000}]
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