0,2

The numbers of domino tilings in A006253, A004003, A006125 give the number of perfect matchings in the relevant graphs. There are results of Jockusch and Ciucu that if a planar graph has a rotational symmetry then the number of perfect matchings is a square or twice a square - this applies to these 3 sequences. - Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Apr 12 2001

Christine.Bessenrodt(AT)Mathematik.Uni-Magdeburg.DE points out that Pachter (1997) shows that a(n) is divisible by 2^n (cf. A065072).

a(n) is the number of different ways to cover a 2n X 2n lattice with 2n^2 dominoes. John and Sachs show that a(n) = 2^n*B(n)^2, where B(n) == n+1 (mod 32) when n is even and B(n) == (-1)^((n-1)/2)*n (mod 32) when n is odd. - Yong Kong (ykong(AT)curagen.com), May 07 2000

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

M. Ciucu, Enumeration of perfect matchings in graphs with reflective symmetry. J. Combin. Theory Ser. A 77 (1997), no. 1, 67-97

S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 406-412.

M. E. Fisher, Statistical mechanics of dimers on a plane lattice, Physical Review, 124 (1961), 1664-1672.

W. Jockusch, Perfect matchings and perfect squares. J. Combin. Theory Ser. A 67 (1994), no. 1, 100-115.

P. E. John and H. Sachs, "On a strange observation in the theory of the dimer problem", Discrete Mathematics, 216 (2000), 211-219.

Pachter, Combinatorial approaches and conjectures ..., Elec. J. Comb. 4 (1997) #R29.

Darko Veljan, KOMBINATORIKA S TEORIJOM GRAFOVA (Croatian) (Combinatorics with Graph Theory) mentions the value 12988816 = 2^4*901^2 for the 8 X 8 case on page 4.

Jaime Rangel-Mondragon, Polyominoes and Related Families, The Mathematica Journal, 9:3 (2005), 609-640.

T. D. Noe, Table of n, a(n) for n=0..25

M. Aanjaneya and S. P. Pal, Faultfree tromino tilings of rectangles

H. Cohn, 2-adic behavior of numbers of domino tilings

S. R. Finch, The Dimer Problem

S. R. Finch, Two Dimensional Monomer Dimer Constant

R. P. Stanley, A combinatorial miscellany

Index entries for sequences related to dominoes

Eric Weisstein's World of Mathematics, Domino Tiling

P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 363

The 36 solutions for the 4 X 4 board, from Neven Juric, May 14 2008:

A01 = {(1,2), (3,4), (5,6), (7,8), (9,10), (11,12), (13,14), (15,16)}

A02 = {(1,2), (3,4), (5,6), (7,11), (9,10), (8,12), (13,14), (15,16)}

A03 = {(1,2), (3,4), (5,9), (6,7), (10,11), (8,12), (13,14), (15,16)}

A04 = {(1,2), (3,4), (5,9), (6,10), (7,8), (11,12), (13,14), (15,16)}

A05 = {(1,2), (3,4), (5,9), (6,10), (7,11), (8,12), (13,14), (15,16)}

A06 = {(1,2), (3,4), (5,6), (7,8), (9,10), (13,14), (11,15), (12,16)}

A07 = {(1,2), (3,4), (5,9), (6,10), (7,8), (11,15), (13,14), (12,16)}

A08 = {(1,2), (3,4), (5,6), (7,8), (9,13), (10,14), (11,12), (15,16)}

A09 = {(1,2), (3,4), (5,6), (7,11), (8,12), (9,13), (10,14), (15,16)}

A10 = {(1,2), (3,4), (5,6), (7,8), (9,13), (10,11), (14,15), (12,16)}

A11 = {(1,2), (3,4), (5,6), (7,8), (9,13), (10,14), (11,15), (12,16)}

A12 = {(1,2), (5,6), (3,7), (4,8), (9,10), (11,12), (13,14), (15,16)}

A13 = {(1,2), (3,7), (4,8), (5,9), (6,10), (11,12), (13,14), (15,16)}

A14 = {(1,2), (5,6), (3,7), (4,8), (9,10), (13,14), (11,15), (12,16)}

A15 = {(1,2), (3,7), (4,8), (6,10), (5,9), (11,15), (12,16), (13,14)}

A16 = {(1,2), (3,7), (4,8), (5,6), (9,13), (10,14), (11,12), (15,16)}

A17 = {(1,2), (3,7), (4,8), (5,6), (9,13), (10,11), (14,15), (12,16)}

A18 = {(1,2), (5,6), (3,7), (4,8), (9,13), (10,14), (11,15), (12,16)}

A19 = {(1,5), (2,6), (3,4), (7,8), (9,10), (11,12), (13,14), (15,16)}

A20 = {(1,5), (2,6), (3,4), (7,11), (8,12), (9,10), (13,14), (15,16)}

A21 = {(1,5), (3,4), (2,6), (9,10), (7,8), (11,15), (13,14), (12,16)}

A22 = {(1,5), (2,6), (3,4), (7,8), (9,13), (10,14), (11,12), (15,16)}

A23 = {(1,5), (2,6), (3,4), (7,11), (8,12), (9,13), (10,14), (15,16)}

A24 = {(1,5), (2,6), (3,4), (7,8), (9,13), (10,11), (14,15), (12,16)}

A25 = {(1,5), (2,6), (3,4), (7,8), (9,13), (10,14), (11,15), (12,16)}

A26 = {(1,5), (2,3), (6,7), (4,8), (9,10), (11,12), (13,14), (15,16)}

A27 = {(1,5), (2,6), (3,7), (4,8), (9,10), (11,12), (13,14), (15,16)}

A28 = {(1,5), (2,3), (6,7), (4,8), (9,10), (11,15), (13,14), (12,16)}

A29 = {(1,5), (2,6), (3,7), (4,8), (9,10), (13,14), (11,15), (12,16)}

A30 = {(1,5), (2,3), (6,7), (4,8), (9,13), (10,14), (11,12), (15,16)}

A31 = {(1,5), (2,6), (3,7), (4,8), (9,13), (10,14), (11,12), (15,16)}

A32 = {(1,5), (2,6), (3,7), (4,8), (9,13), (10,14), (11,15), (12,16)}

A33 = {(1,5), (2,6), (3,7), (4,8), (9,13), (10,11), (14,15), (12,16)}

A34 = {(1,5), (2,3), (4,8), (6,10), (7,11), (9,13), (14,15), (12,16)}

A35 = {(1,5), (2,3), (6,7), (4,8), (9,13), (10,14), (11,15), (12,16)}

A36 = {(1,5), (2,3), (6,7), (4,8), (9,13), (10,11), (14,15), (12,16)}

f := n->2^(2*n^2)*product(product(cos(i*Pi/(2*n+1))^2+cos(j*Pi/(2*n+1))^2, j=1..n), i=1..n); for k from 0 to 12 do round(evalf(f(k), 300)) od;

Cf. A028420, A006253, A006125, A065072, A124997.

Main diagonal of array A099390.

Sequence in context: A001070 A095229 A047832 this_sequence A060739 A134366 A127234

Adjacent sequences: A004000 A004001 A004002 this_sequence A004004 A004005 A004006

nonn,easy,nice

N. J. A. Sloane (njas(AT)research.att.com), Greg Huber

Corrected and extended by David G. Radcliffe (radcl008(AT)umn.edu)