Looks difficult but is simple

Algebra Level 3

Let S n = k = 1 4 n ( 1 ) k ( k + 1 ) 2 k 2 \large\displaystyle S_n = \sum_{k=1}^{4n} (-1)^{\frac{k(k+1)}2} k^2 , then which of the following is a possible value of S n S_n ?

1066 1038 1056

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