An analytical recurrence formula for the zeroth-order metal wire plasmon of terahertz waves is presented. In the derivation, the property that the relative permittivity of a metal in the spectral region of terahertz wave is huge, the property that the effective index is about 1, some properties of modified Bessel functions, and a suitable Taylor expansion are employed. The recurrence formula is numerically tested for 11 nonmagnetic metals for the whole spectral region of terahertz waves and for the wide radius range from $10\mu \mathrm{m}$ to infinity. We find that the relative deviation for the effective index always becomes smaller than $2.1\times {10}^{-6}$ after only four recurrences if a suitable initial input is used. Some related problems, such as the connection to the Newton method and the dependence of the accuracy on frequency, are also discussed.