Pallindromically, what is 111111111 x 111111111?

111,111,111 x 111,111,111 = 12345678987654321 from WTF

$$\left(\sum_{k=0}^n 10^k\right)^2 = \sum_{i=0}^n \sum_{j=0}^n 10^{i+j} $$ There are $k+1$ occurrences of $10^k$, namely $(i,j)=(0,k),(1,k-1),\ldots, (k,0)$, if $k \le n$, and $2n+1-k$, namely $(i,j) = (k-n,n),(k-n+1,n-1),\ldots,(n,k-n)$, if $n < k \le 2n$. If $n \le 8$ this means the decimal representation of $(1\ldots1)\times(1\ldots1)$ is $12 \ldots n (n+1) n \ldots 21$.