Square of Repunit times Sum of Digits

Theorem
The following pattern emerges:

and so on, up until $999 \, 999 \, 999^2$ after which the pattern breaks down.

Proof
From Square of Repunit:

and so on.

Then from 1+2+...+n+(n-1)+...+1 = n^2:

and so on.

Then:

The pattern breaks down after $9$:


 * $1 \, 111 \, 111 \, 111^2 = 1 \, 234 \, 567 \, 900 \, 987 \, 654 \, 321$