OK, perhaps it’s a roundabout way to get there, but here it is…

Original sequence: 4,20,56,120,220,…

Identify the sequence pattern: 4(16=),20(36=),56(64=),120(100=),220,… I do realize these are not actual mathematical equations, I’m just trying to illustrate the pattern.

Each number that is added to the previous number to get the next is a perfect square.

When reduced to it’s square root, a simpler pattern is revealed: 4,6,8,10,…

The next number in this simple sequence is obviously 12.
12 squared is 144. Therefore:

4(16=),20(36=),56(64=),120(100=),220(+144=),364

364 is the next number in the sequence.

Sorry, all of my addition symbols were stripped out of my previous post – don’t really know why. The patterns should read:

4(plus16=),20(plus36=),56(plus64=),120(plus100=),220

and

4(plus16=),20(plus36=),56(plus64=),120(plus100=),220(plus144=),364

The closest I came up with was USHER with 5 letters and 4 pronouns :(
US
HE
SHE
HER