A lot of values for $n$ !

$n^2+6n+646=k^2$ $(n+3)^2 + 637 = k^2$ $k^2 - (n+3)^2 = 637$ $(k-n-3)(k+n+3) = 7 \times 7 \times 13$

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$k-n-3=1, \; k+n+3=637 \Rightarrow n = 315$ $k-n-3=7 \wedge k+n+3=91 \Rightarrow n = 39$ $k-n-3=13 \wedge k+n+3=49 \Rightarrow n = 15$

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$k-n-3 > k+n+3 \Rightarrow n < 0$

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$\boxed{\sum n = 369.}$