I broke my EU2000i in on Honda 10W-30 oil. They filled it when I bought it and gave me a quart of it.
I changed the oil after 30 minutes of running to get the initial break-in glitter out of the engine. Then I changed every 2-5 hours until the drained oil came out clean (without any metal glitter when viewed in the sunshine).
I'm now using up the last of a jug of Mobil 1 10W-30 oil in it. I have a new jug of Mobil 1 Turbo Diesel Truck 5W-40 I got on sale in the queue for the next oil change. Honestly, the 10W-30 did just fine in the winter in SE Michigan this last year during four power outages. It pulled over and started up just fine in 20 degree weather.
Now I'm on the 100hr or annual OCI plan. (Actually, I'll only change it at the end of the season if it has more than 20 or so hours on it. If not, I'll leave the oil in for another year. My annual use varies depending on camping sites and power outages at home.)