When Is the Digital Root Three?

For how many positive integers n1000 n \leq 1000 , is the digit root of n2+n+1 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 312+31+1=99331^2+31+1 = 993 would be 3 since 9+9+3=21,2+1=39 + 9 + 3 = 21, 2+1 = 3 .

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