Does sketching the Pascal Triangle help?

Let

\[\displaystyle F(n) = \sum_{i=0}^{\lfloor \frac{n}{2} \rfloor } {{n-i}\choose i}.\]

What is the value of \(\text{gcd}(F(101), F(118))\)?

\[\] Notation: \(\gcd(\cdot) \) denotes the greatest common divisor function.

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