According to this wiki :
performing a series of three Auto Attacks makes the amount of damage of each attack increase
I remember that from the tutorial at the beginning of the game plus the fact that if you use an art after the third auto attack it will deal more damage.
For the art to deal extra damage you have to press the button right when the auto attack hits. Then there is a small white circle that pops when you do it properly if I remember correctly. If you then chain with another art after the previous one hit you have the same effect (the white circle popping). You can also continue by using a blade special with the correct timing.
So basically if I were to perform the following chain :
auto attack > auto attack > auto attack > art 1 > art 2 > art 3 > blade special
What would the damage bonus be at each stage ? Is art 1 the only one which benefits from the third auto attack bonus ?