We prove a general theorem to bound the total variation distance between the distribution of an integer valued random variable of interest and an appropriate discretized normal distribution. We apply the theorem to $2$-runs in a sequence of i.i.d. Bernoulli random variables, the number of vertices with a given degree in the Erdös–Rényi random graph, and the uniform multinomial occupancy model.
"Discretized normal approximation by Stein’s method." Bernoulli 20 (3) 1404 - 1431, August 2014. https://doi.org/10.3150/13-BEJ527