^(?=.{8,20}$)(?![_.])(?!.*[_.]{2})[a-zA-Z0-9._]+(?<![_.])$

^(?=.{8,20}$)(?![_.])(?!.*[_.]{2})[a-zA-Z0-9._]+(?<![_.])$
 └─────┬────┘└───┬──┘└─────┬─────┘└─────┬─────┘ └───┬───┘
       │         │         │            │           no _ or . at the end
       │         │         │            │
       │         │         │            allowed characters
       │         │         │
       │         │         no __ or _. or ._ or .. inside
       │         │
       │         no _ or . at the beginning
       │
       username is 8-20 characters long


If your browser raises an error due to lack of negative look-behind support, use the following alternative pattern:

^(?=[a-zA-Z0-9._]{8,20}$)(?!.*[_.]{2})[^_.].*[^_.]$