sb = {1, 1};
Do[
 sb = sb~Join~{Total@sb[[i - 1 ;; i]], sb[[i]]}
 ,
 {i, 2, 1000}
 ]
Take[sb, 15]
Flatten[FirstPosition[sb, #] & /@ Range[10]]
First@FirstPosition[sb, 100]
AllTrue[Partition[Take[sb, 1000], 2, 1], Apply[GCD] /* EqualTo[1]]