<s>[INST] Find the exact closed from expression of $1^2 + 3^2 + 5^2 + · · · + (2n + 1)^ 2$ I know the above expression equals to $\frac{n(2n−1)(2n+1)}{3}$, but how exactly can i come up with something from scratch? 
 [/INST] $$(2k-1)^2=8\binom{k}{2}+1$$
but since
$$\sum_{k=1}^{N}\binom{k}{2}=\binom{N+1}{3}$$
it follows that:
$$\sum_{k=1}^{N}(2k-1)^2 = N+8\sum_{k=1}^{N}\binom{k}{2}=N+8\binom{N+1}{3}=\frac{1}{3}N(2N+1)(2N-1).$$
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