Look like a flower but be the serpent underneath

Take a look at this:

\[\begin{matrix} 1.1 & 1.2 & 1.3 & 1.4 & 1.5 & ... \\ 2.1 & 2.2 & 2.3 & 2.4 & 2.5 & ... \\ 3.1 & 3.2 & 3.3 & 3.4 & 3.5 & ... \\ 4.1 & 4.2 & 4.3 & 4.4 & 4.5 & ... \\ 5.1 & 5.2 & 5.3 & 5.4 & 5.5 & ... \\ ... & ... & ... & ... & ... & ... \end{matrix}\]

Now, let

\[{ a }_{ 1 }=1.1,\quad { a }_{ 2 }=2.1,\quad { a }_{ 3 }=1.2\\ { a }_{ 4 }=3.1,\quad { a }_{ 5 }=2.2,\quad { a }_{ 6 }=1.3\\ { a }_{ 7 }=4.1,\quad { a }_{ 8 }=3.2,\quad { a }_{ 9 }=2.3\\ ...\]

\[\sum _{ k=\frac { n(n-1) }{ 2 } +1 }^{ \frac { (n)(n+1) }{ 2 } }{ { a }_{ k } } =S(n)\]

\[0<n\le 2015\]

Find the sum of all the values of integer \(n\) such that \(S(n)\) is an integer.


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