I've been trying and trying... but there does not seem to be three

Number Theory Level 5

Consider the expression \(\sqrt {\frac {(n+1)(2n+1)}{6}}\).

For some whole numbers \(n\), the expression is also a whole number. Let the three smallest such whole numbers be \(a_{1}, a_{2}, a_{3}\). Find the value of \(\sum_{i=1}^3 a_{i}\).

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