The sum of all three digited numbers that can be formed from the digits 1 to 9 and when the middle digit is perfect square 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 

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

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

When repetitions are not allowed

${ }^7{p}_1\left(101\right)\left(\sum 9-1\right){+}^8{p}_2\times 10$

${ }^7{p}_1\left(101\right)\left(\sum 9-4\right){+}^8{p}_2\times 40+$${ }^7{p}_1\left(101\right)\left(\sum 9-9\right){+}^8{p}_2\times 90=93,387$