A summoning of sums

For positive integers \(n\) let \(S(n)\) be the number of positive integer pairs \((a,b)\) such that \(a^{2} + a + n = b^{2}\).

Let \(n_{k}\) be the least positive integer for which \(S(n_{k}) = k\).

Find \(\displaystyle\sum_{k=0}^{4} n_{k}\).

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