I am using the NAD 1983 2011 UTM Zone 13N system for my data in ArcGIS. I really want to know how the equations ArcGIS used to calculate the X and Y.
I suggest you have a look at Esri's page on Geographic Transformation Methods. In it, they explain that
A geographic transformation always converts geographic (latitude–longitude) coordinates. Some methods convert the geographic coordinates to geocentric (X,Y,Z) coordinates, transform the X,Y,Z coordinates, and convert the new values back to geographic coordinates.
Equation Based Methods describes four methods for converting coordinates:
This page on grid based methods is also likely relevant.
Also have a look at this technical article How To: Determine which NAD_1983_To_WGS_1984 transformation to use.
I'm unsure which method above was used for your specific calculations.