Ugly Fibonacci Sums

Algebra Level 4

If $$k$$ is an integer that satisfies the identity $\left(\sum_{i=1}^{n}F_{2i-1}\right)^2-\left(2\sum_{i=1}^{n}F_{2i-1}^2\right)=\left(\sum_{i=1}^{2n-2}(-1)^iF_i^2\right)-k$ then find $$k$$.

$$\text{Details and Assumptions}$$

$$F_1=F_2=1$$, and $$F_n=F_{n-1}+F_{n-2}$$.

[You should assume that $$n \geq 2$$ for the summations to make sense.]

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