We report the results from Mayer-sampling Monte Carlo calculations of the virial coefficients of the united-atom TraPPE-UA model of normal alkanes. For alkane chain lengths from n=2 to 20 (where n is the number of carbon atoms), results are given for the virial coefficients B(2), B(3), and B(4); results for B(5) are given for chains up to length n=12; and results for B(6) are given for chains of length n=2, 3, and 4. In all cases, values are given for temperatures ranging from 200 K to 2000 K in 20-50 K increments. The values are used to calculate the equation of state for butane and the pressure-density behavior is compared to experimental data at 350 and 550 K. Critical points are calculated for all systems and compared to simulation data previously taken for the same molecular model, and to experiment. The comparison with temperature is very good (within 1.5% for all chain lengths up to n=12), while the critical density is underestimated by about 5%-15% and the critical pressure is given within about 10%. The convergence behavior of the virial equation of state as applied across the n-alkane series is well characterized by corresponding states, meaning that the accuracy at a given density relative to the critical density does not deteriorate with increasing chain length.