Assuming you want 1-based coordinates, then you would instead want:

frame 1: nuc_start = 3*(prot_start-1)+1
frame 2: nuc_start = 3*(prot_start-1)+2
frame 3: nuc_start = 3*(prot_start-1)+3

This assumes that you simply ignore the first 2 bases when translating frame 3, rather than padding the translated sequence with something.

It works the same for the other strands, how you do the math will depend on whether you reversed the sequence or not. If you reversed things, then the resulting coordinates will be from the end of the chromosome/contig. Also, frame -1 will be dealt with the same as frame +3 and -3 the same as +1.

If this is unclear, draw out a short sequence, how this works should be apparent then.

Edit: And as Pierre pointed out, this all assumes that there's no splicing...