(Logic) Complicated formula-autofill problem

Hi everyone, apologies if this is a problem that can be solved via a solution found elsewhere in the forums. I've been searching a while and haven't yet come up with anything. I'm pretty new to using excel, so bear with me if I use completely wrong terminology. I'll try to be succinct and as clear as possible

Is there a way to copy a formula from one cell and have the information progress on a different scale than what the actual grids would imply?

For example, I have a table using the Vlookup function to populate a field with a list of over 400 names. I have a separate table for each of these names, which will hold other information relative to that name, that has to be laid out in a specific way (I can't make all the information correspond on the x and y axis to the lookup tables). The cells also need to be relative, so that if I change a Name in my Vlookup, it will change on the corresponding sheet. None of the information requires math, only copying information organized elsewhere.
So, I need to copy a formula from B3 to B19, but the Vlookup number needs to only move down 1 name, not 16...

I know that I can copy the formula with the cells organized so that they are immediately next to each other (By using the sort function and a numbered column corresponding to each space that needs to have the name), and the formula will do what I need it to... but I need to be able to create more of these blocks relatively quickly, and I have about 20 different pieces of information that are also relative to my lookup table but need to be in different locations.

Obviously I can manually enter each formula, or reorganize the entire sheet every time I need to copy formulas, but I'm looking to be able to quickly create more blocks of information where the formulas each change only one step down the look-up table, even though they are being represented 16 cells downward each time in the final sheet.

Any suggestions?