The sequence is:
4,7,7,0,1,1,4,8,3,2,3,9,4,1,5,3,0,6,4,5,0,5,1,0,9,1,2,7,7,1,4,9,5,5,6,9,5,1,?
It's very simple indeed you won't need to do any calculation, but you may need to do a little one, good luck. Yes you may really need to read this hint.
Hint1:
Three things make up it, one of them is the first of Good Boy. A cat tamed the boy. The third one is an acronym relating to computer networks.
Hint2:
It is neither due to Euler, nor by Apery.
Hint3:[New]
Some constant perhaps?
Edit: Hint edited.
Edit2: Ok I may add other hints if you comment something (maybe the problem you're facing or which type of hint should be there).