The best oil to use is a LAWNMOWER OIL synthetic, semi-synthetic , strait mineral makes very little diffrence so long as the viscosity is withing those listed in the mower manual.
Running synthetic oil will not hurt your engine no will it make the engine run for longer between services or any better .
Thus there is zero advantage in using it .
But if you want to then go ahead but do an oil change after the first mow to get rid of the last of the previous oil.
To get the best out of your mower, change the oi on the last mow of the season, the instant you finish mowing so all of the crub is in suspension and you then have clean oil, with no acid by-products in it sitting in your mower during the off season.