For the year $2025$

$$2025=(20+25{)}^2$$

Given positive integers $a$ and $b$, the concatenation $ab$ we call a $2025$-number if $ab=(a+b{)}^2$.
Other examples are $3025$ and $81$.
Note $9801$ is not a $2025$-number because the concatenation of $98$ and $1$ is $981$.

Let $T(n)$ be the sum of all $2025$-numbers with $n$ digits or less. You are given $T(4)=5131$.

Find $T(16)$.