This is an extension of GCD of Fibonacci.

Denote the nth term of the Fibonacci sequence as F(n), with F(0)=0 and F(1)=1. Let S_n be the set {F(n), F(n+1), F(n+2), F(n+3)}.

For what values n does there exist a pair of numbers from S_n with a GCD greater than 1?