The formula JyZude mentions is referred to as Binet's Formula for calculating the nth Fibonacci number. Of the two references I've read which skimmed across this subject, both have noted that Binet probably wasn't the first to discover this formula, and found it on his own 100 years after. They differ, in that one claims de Moivre discovered it first, and the other claims Euler did. Binet, however, eventually wound up getting the credit for it.