So I have been wanting to know what the clock reset password formula for these games. I know there are a ton of calculators I could use online in order to get the password for my game, but I don't see the formula used in these calculators anywhere. Does anyone know where I can find the formula used in the game to generate the password (I only really care about the formula for the US versions of Gold and Silver)?

The formula (which is the same in Gold/Silver and Crystal) is well-explained at the top of the JavaScript code in this calculator (here is a direct link to the JavaScript file if that's easier to read). The character values for the trainer name are a bit below the explanation in the calculator, and are represented by the "tile number" associated with that character.

There are a few missing valid characters from that calculator, in case they are in the first 5 characters of your trainer name:

​​⠀
​​⠀= 127 (space)
( = 154
) = 155
: = 156
; = 157
[ = 158
] = 159
<PK> = 225 (1st half of stylized Pokémon symbol)
<MN> = 226 (2nd half of stylized Pokémon symbol)
- = 227
? = 230
! = 231
. = 232
× = 241 (multiplication symbol)
/ = 243
, = 244