Variance of the range of a random walk

T_n, the expectation of the square of the number of distinct sites occupied by a random walk in steps 1 throughn, is obtained from its relation to the dual first occupancy probabilityF_ij(x, x), and the latter quantity is obtained from a recursion with the first occupancy probabilityF_k(x). The varianceV_n of the number of distinct sites occupied is calculated directly from T_n; the procedure is illustrated by the calculation ofV_n (4096 /n) and the derivation of asymptotic expansions forV_n for a particular random walk in dimensions 1 through 3.Work completed under the auspices of the United States Department of Energy.