During production of our cannon ball number video, Matt supplied this extra information:

For the bigger numbers I checked until 10^11. Here is what I found (back in 2013)!

There are four triangle cannonball numbers below 10^9: 10, 120, 1540, 7140

There is one square cannonball number below 10^9: 4900

There is one pentagon cannonball number below 10^9: 946

There are two octagon cannonball numbers below 10^9: 1045, 5985

There is one decogon cannonball number below 10^9: 175

There is one 11-agon cannonball number below 10^9: 23725

There are two 14-agon cannonball numbers below 10^9: 441, 195661

There is one 17-agon cannonball number below 10^9: 975061

There is one 20-agon cannonball number below 10^9: 3578401

There is one 23-agon cannonball number below 10^9: 10680265

There is one 26-agon cannonball number below 10^9: 27453385

There is one 29-agon cannonball number below 10^9: 63016921

There is one 30-agon cannonball number below 10^9: 23001

There is one 32-agon cannonball number below 10^9: 132361021

There is one 35-agon cannonball number below 10^9: 258815701

There is one 38-agon cannonball number below 10^9: 477132085

There are two 41-agon cannonball numbers below 10^9: 55202400, 837244045

There is one 43-agon cannonball number below 10^9: 245905

There is one 50-agon cannonball number below 10^9: 314755

There is one 88-agon cannonball number below 10^9: 48280

There is one 145-agon cannonball number below 10^9: 101337426

There is one 276-agon cannonball number below 10^9: 801801

There is one 322-agon cannonball number below 10^9: 1169686

There is one 823-agon cannonball number below 10^11: 197427385

There is one 2378-agon cannonball number below 10^11: 432684460

There is one 2386-agon cannonball number below 10^11: 29437553530

There is one 31265-agon cannonball number below 10^11: 90525801730