Find the sum of $-1^2-2^2+3^2+4^2-5^2-6^2+\cdots$

HINT: Split your last sum into the sum of two arithmetic progressions, each of length $n$.

An alternative is to calculate a few values of the sum, guess a closed form, and then prove the closed form. For $n=1,2,3$ the sum in question is $20,72,156$, respectively. Note that $20=4\cdot5$, $72=8\cdot9$, and $156=12\cdot 13$.

Hint: except $-1^2$ make pairs of other terms: $$-1^2+(-2^2+3^2)+(4^2-5^2)+(-6^2+7^2)+....$$ and then proceed further to get $-1^2+2+3+4+5+6+...$ and than solve it to get the result

$ =2[4+6+12+14+20+22+\cdots+(8n-4)+(8n-2)] \\ = 4[(2+3)+(6+7)+(10+11)+\cdots+(8n-3)] \\ = 4[5+13+21+\cdots+(8n-3)] \\ = 4 \sum_{k=1}^n (8k-3) \\ = 4 (4n^2+n) \\ = 16n^2+4n $