Three digit numbers in which the middle one is a perfect square are formed using the digits 1 to 9. Their sum is:

The value of the number xyz is 100x + 10y + z, where as x and z can be any two of the digits 1 to 9. But y can only be any one of the digits 1, 4, 9.
$\therefore$ Total value of the 3 digit numbers so formed is 

$\mathrm{V}=100(1+2+3+\dots .+9)\cdot 9\cdot 3+10(1+4+9)\cdot 9\cdot 9+(1+2+\dots .+)\cdot 9\cdot 3$

$\therefore \mathrm{V}=121500+11340+1215=134055$