Perfect Squares Divisible by Repeating 1 Digits

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Are there any perfect squares with the form $$\frac{A(10)^n+B}{\sum_{k=0}^{n-1} 10^k},$$ or, in other words, is $$\sqrt{\frac{A(10)^n+B}{\sum_{k=0}^{n-1} 10^k}}$$ ever a natural number?