This is a map of population density, made using Jonathan Schroeder's county-level decadal estimates. I've interpolated populations for intermediate years by interpolating cubic splines to log-density; essentially, that means that it assumes a smooth change in the rate of growth for each county over time. This is nice because the continuous filling makes it easier, for me at least, to see the process of settlement as continuous.
Hover your mouse or drag your figure over the map to zoom around: press the year buttons to start the video at a year.
On the other hand: this produces wrong information! For instance, many Oklahoma counties became heavily populated in 1891 or 1893 as soon as the federal government; but the algorithm doesn't fill them in until later. Speaking of Oklahoma, this is census population, which does not include most native nations.