Consider the sequence $T_n$ formed by combining $n$ and $n^2$ into one number. ie. (A053061) $$T_n=\{11,24,39,416,525,636,749 \cdots\}$$ It is easy to see $$T_n= 10^{\lceil 2 \log_{10}(n) \rceil } n+ n^2$$
I looked at the sequence closely trying to find if there are any perfect squares in the sequence but wasn't able to upto $n=100$. I also was able to prove that if $n^{th}$ term is a perfect square then :
1) $n \equiv 8 \text{ or } 0 (\text{ mod } 9)$
2)In the case where $n=9m$ , m is not a square free number.
But I am unable to attack the question
Do there exist any perfect square in the sequence?