# 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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