Squares whose Digits form Consecutive Increasing Integers

Theorem
The sequence of integers whose squares have a decimal representation consisting of the concatenation of $2$ consecutive integers begins:
 * $428, 573, 727, 846, 7810, 36 \, 365, 63 \, 636, 326 \, 734, \ldots$

Proof
We have:

They can be determined by inspection.