A347054 Number of domino tilings of a 32 X n rectangle.

1, 1, 3524578, 1117014753, 170220478472105, 224916047725262248, 12348080425980866090537, 30648981125778378496845537, 1010618564986361239515088848178, 3596059736380751648485086101179655, 87171995375835553001398855677616476448, 391978133958466896956216157693001644153072

Offset: 0

Author

A. M. Magomedov and Serge Lawrencenko, Aug 14 2021

Keywords

• nonn

Comments

It is known that the number of domino tilings of an m X n rectangle is equal to the number of perfect matchings in the m X n grid graph.

References

• 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

Links

• 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.

Crossrefs

Cf. A099390, A340532.

Column n=32 of A187596.

Programs

Mathematica

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}]

Formula

a(n) = Product_{j=1..16} (Product_{k=1..floor(n/2)}(4*(cos(j*Pi/33))^2+ 4*(cos(k*Pi/(n+1)))^2)) (special case of the double product formula in A099390).