On the Sequence enuerating Monomer-Dimer Tilings of an , 2,  by  , 2 n,

     rectangle

                           with Monomer Density, 1/2



Definition: Let s(n) be the number of Monomer-Dimer Tilings of an , 2,  by  ,

    2 n,  rectangle with monomer density, 1/2

          in other words, there are , 2 n, monomers and , n, dimers .

                           The first, 40, terms are



[4, 29, 234, 1982, 17266, 153190, 1376770, 12492245, 114187660, 1049883133,

    9699127442, 89958450168, 837142603118, 7812580075084, 73090224445226,

    685264230399038, 6436937423661874, 60566198366788934, 570733434816490002,

    5385453494993166648, 50879254856162770278, 481215960411301946464,

    4555938590521017527594, 43173507650928728396122, 409473681834988404005746,

    3886647642539955768448190, 36918105787794853599345522,

    350910215148472208648638832, 3337525855931869660050973966,

    31761939199222935520931198124, 302431221897478581988762556682,

    2881167714769772358226784808405, 27461179282353633657132636191620,

    261858206520098121168998194083685, 2498033493426418424939090815809866,

    23839933317681498984351847203841034, 227602496009742790996855400041479138,

    2173729295836480637102448712389487998,

    20767407121932693511913432103600399714,

    198472349624607055189750926412824112813]

                                                                     n
                                                          9.676287316
         The asymptotics of the sequence, numerically is, ------------
                                                               1/2
                                                              n



                                  Theorem: Let

                                   infinity
                                    -----
                                     \            n
                            f(x) =    )     s(n) x
                                     /
                                    -----
                                    n = 0



             Then f(x)=P satisfies the following algebraic equation



                                                          2
2 x                          2      4 (23 x + 1) (x - 1) P
--- + (1/27 + 10/9 x + 1/27 x ) P - -----------------------
27                                            27

       /         76            2   140  3    4\  4
     + |- 5/27 - -- x + 406/9 x  - --- x  + x | P  = 0
       \         27                27         /



                              and in Maple format



2/27*x+(1/27+10/9*x+1/27*x^2)*P-4/27*(23*x+1)*(x-1)*P^2+(-5/27-76/27*x+406/9*x^
2-140/27*x^3+x^4)*P^4 = 0


    The exact value of the connective constant, derived numerically above is

                                       1
           ----------------------------------------------------------
                       4         3          2
           RootOf(27 _Z  - 140 _Z  + 1218 _Z  - 76 _Z - 5, index = 1)



                             and in floating-point



                             9.6778602034987359177



             This ends this article, that took, 2.124, to generate.