  Counting the number of N-step self-avoiding walks (SAWs) on a lattice is one
of the most difficult problems of enumerative combinatorics. Once we give up
calculating the exact number of them, however, we have a chance to apply
powerful computational methods of statistical mechanics to this problem. In
this paper, we develop a statistical enumeration method for SAWs using the
multicanonical Monte Carlo method. A key part of this method is to expand the
configuration space of SAWs to random walks, the exact number of which is
known. Using this method, we estimate a number of N-step SAWs on a square
lattice, c_N, up to N=256. The value of c_256 is 5.6(1)*10^108 (the number in
the parentheses is the statistical error of the last digit) and this is larger
than one googol (10^100).
