The Written Digits Sequence
Perl 6, 45 bytes

{({[+] @_.join.uninames&gt;&gt;.comb X-6}...*)[$_]}

Try it online!
No need for fancy moduloing when you can get the name of the digit directly! Anonymous code block that returns the nth value of the sequence, or you can pass in a range to get a list of values
Explanation:
{(                                     )[$_]}  # Index input into:
  {                               }...*        # An infinite sequence
                                               # Where each element is
   [+]   # The sum of
       @_.join  # All previous elements joined together
              .uninames  # The unicode names for each character
                         # These are names in the form &quot;DIGIT ONE&quot;
                       &gt;&gt;.comb  # Split each to lists of characters
                               X-6  # Subtract 6 from each