Love, the Fibonacci way

Remember the Fibonacci sequence?  Start with 1.  Then another 1.  Now each subsequent number is the sum of the two previous numbers, so we have 1, 1, 2, 3, 5, 8, 13, … forever!  This sequence allegedly began with Fibonacci’s modeling of a rabbit population. 

Bonus: can you extend the sequence the other direction – what number would come before the first 1?  What family of function would you guess the sequence to be?