Find the logic beyond the summands in the above equation
and derive the values of n and S_n .

Oddly enough I did consider something along these lines.

Unfortunately the problem then doesn't work because the (unedited) sequence (S(40) to S(47) runs 4606, 250, 765, 1300, 1855, 2430, 3025, 3640, but S(46), being 55*55, is a unique term, for obvious reasons.

I therefore assumed that the +...+ had been transposed from between 4606 and 3640 to immediately after those terms, where its current position looks a bit odd. Anything else would indeed be 'duplicity' rather than 'duplication', which is what I suspect you mean here.