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We devise explicit partitioned numerical methods for second–order-in-time scalar stochastic differential equations, using one Gaussian random variable per timestep. The construction proceeds by analysis of the stationary density in the case of constant-coefficient linear equations, imposing exact stationary statistics in the position variable and absence of correlation between position and velocity; the remaining error is in the velocity variable. A new two-stage “reverse leapfrog” method has good properties in the position variable and is symplectic in the limit of zero damping. Explicit new “Runge–Kutta leapfrog” methods are constructed, sharing the property that $q_{n+1}=q_n+\frac{1}{2}(p_n+p_{n+1})\Delta t$, whose mean-square velocity order increases with the number of stages.