Are you aware what the source of H-H splitting is? If yes skip to the next paragraph. Basically a hydrogen atom can have its spin aligned with the external magnetic field (and producing a signal downfield on the adjacent hydrogen I believe) or have its spin aligned against the field. For reasons I'm not sure of either alignment is equally likely. So if there is no adjacent hydrogen you get a singlet because the hydrogen is in its own environment. If there is 1 adjacent hydrogen its spin can be aligned with or against the external magnetic field, and hence produces a double, both of equal height. If there are 2 adjacent hydrogens, they can be both with, both against, or cancel each other out with regard to the magnetic field, so you get a triplet with the middle peak twice as large as the other two peaks. With 3 adjacent hydrogens, you have 4 possibilities and so on.
Now when you have hydrogens on both sides that are in different chemical environments, and lets say you have two on either side of the reference hydrogen (lets call it Hr) if both the hydrogens on one side are supporting the field, and the two hydrogens on the other side are against the field they don't cancel each other out because they are in different environments (whereas if they were equivalent they would cancel each other out), so you have to multiply the N+1s