The F1 engine [1] (Saturn V, first stage) used another interesting way to improve efficiency with a gas generator cycle: using the turbopump exhaust gas as a cooling film in the engine nozzle. The fuel-rich exhaust was relatively cool compared to the flame generated by the rocket engine itself, and thus protected the nozzle from the most intense heat.

This is why, close up, the flame looks almost black close to the nozzle [2].

[1] https://en.wikipedia.org/wiki/Rocketdyne_F-1

[2] https://upload.wikimedia.org/wikipedia/commons/7/71/F-1_Engi...

Edit: Hopefully clarified a little, and changed the link in [2] (from https://youtu.be/DKtVpvzUF1Y?t=125).