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:<math>\begin{align}
\sigma(n) &= \sigma(n-1)+\sigma(n-2)-\sigma(n-5)-\sigma(n-7)+\sigma(n-12)+\sigma(n-15)+ \cdots \\[12mu]
&= \sum_{i\in\N} (-1)^{i+1}\left( \sigma \left( n-\frac{1}{2} \left( 3i^2-i \right) \right) + \sigma \left( n-\frac{1}{2} \left( 3i^2+i \right) \right) \right),
\end{align}</math>
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