When Is the Digital Root Three?

For how many positive integers \( n \leq 1000 \), is the digit root of \( n^2 + n +1 \) equal to 3?

Details and assumptions

The digit root of a number is a single digit value obtained by iterative digit sums. For example, the digit root of \(31^2+31+1 = 993\) would be 3 since \(9 + 9 + 3 = 21, 2+1 = 3 \).

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