In the numbers below, they mostly show insertion of zeros. But the set begining with 81 runs a bit differently by inserting various digits. And it appears 81*9 =729 contiunes with the insertion of digits. I didn't pursue this one further.
I'm sure those of you that are pattern finders will see also that the sum of the digits reduce to 9.
Also, why don't ALL numbers that reduce to 9 give these "unusual" sequences? For example, 27 doesn't produce a very "pleasing" sequences of digits.
9, 99, 999, 9999, ...
18, 1818, 181818, ...
45, 4545, 454545, ...
36, 3636, 363636, ...
81, 8181, 818181, 81818181, ...
108, 108108, 108108108, ...
81*9 = 729
DrDave