Does sketching the Pascal Triangle help?

Probability Level 5

Let

F(n)=i=0n2(nii).\displaystyle F(n) = \sum_{i=0}^{\lfloor \frac{n}{2} \rfloor } {{n-i}\choose i}.

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

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

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