Can I use the seamless for the liner, then add 1/2 inch wall over that, and another 1/2 inch wall over that. In other words, I would be using 3 pipes, stepping down toward the muzzle...
OK, to address the question directly. First, you should not use any "pipe" in the construction of any cannons; pipe is designed for water and only the smallest diameters are rated over 1000 psi, which is way too little for a cannon.
Next, the Parrott rifle was made by shrink fitting a reinforcement over a cast iron tube. It worked well although the typical failure mode was to burst just in front of the reinforcement, which says to me that the reinforcement should have been longer. You could make a barrel by layering several tubes to get the thickness required in the places that need the strength. A lot of British barrels were made in this fashion.
The point of highest stress is around the powder chamber and a little in front of it and that is where the wall thickness = bore diameter recommendation comes into play. Once forward of that point, the barrel can get thinner as is typically the case with professionally designed barrels.
When the additional diameter is created from additional tubes, the tubes have to fit closely together; an interference fit is best as the compression of the inner tube has to be reversed by the chamber pressure before the tensile strength of the inner tube comes into play. If the tubes do not fit closely together, then the inner one will expand until it contacts the outer one. Too much expansion will cause the inner tube to crack which will transfer the pressure to the outer tube which may burst then or burst on a subsequent shot.
You have proposed to use 9/16" wall tube for the inner tube and add a layer of 1/2" wall tube over that for some distance and add another layer of 1/2" tube over the first two tubes. This would give a wall thickness of 1 9/16" (1.562") compared to the recommended 1.72". For a chamber pressure of 20000 psi, the inner tube would see a stress of about 26000 psi which is well within the strength of a steel tube. If the wall thickness were 1.72", the stress number would be 25000 psi which is not much less. Note the these numbers represent only one of a number of points that need to be resolved during the design and construction of a cannon barrel.
The critical issue with the layered tube, however, is making the fits between the tubes be tight. Otherwise the inner tubes will expand until they touch the outer tubes and the stresses will be much higher.
