Maximum hydrogen atom bonds per a single carbon atom
I would look at it like recursive problem. Starting with the simplest situation:

Carbon has 4 electrons available to form covalent chemical bonds (so 4 simple bonds for example). Hydrogen has one electron - 1 bond.

If you add one more carbon to the situation, you have to bond it somewhere - you need to use the first carbon. Now with this new carbon you added 3 more possible places to add 3 more elements (like hydrogens) but you canceled one hydrogen place on the first carbon. So every time you add another carbon, you can go up only for 2 hydrogen. It is actually called homologous addition or something like that. (Sorry for my english as well)

I ignored the oxygen whole time, cos as far as he is 2 bonded, he never adds anything he either substracts two hydrogen places, or you can instert it between a carbon and a hydrogen.