Dimer-monomer model on the Sierpinski gasket

We present the numbers of dimer-monomers M_d (n) on the Sierpinski gasket S G_d (n) at stage n with dimension d equal to two, three and four. The upper and lower bounds for the asymptotic growth constant, defined as z_{S Gd} = lim_{v → ∞} ln M_d (n) / v where v is the number of vertices on S G_d (n), are derived in terms of the results at a certain stage. As the difference between these bounds converges quickly to zero as the calculated stage increases, the numerical value of z_{S Gd} can be evaluated with more than a hundred significant figures accurate. From the results for d = 2, 3, 4, we conjecture the upper and lower bounds of z_{S Gd} for general dimension. The corresponding results on the generalized Sierpinski gasket S G_{d, b} (n) with d = 2 and b = 3, 4 are also obtained.