Where is the middle of a Fibonacci sequence?

Abstract

As one might expect, a generalized Fibonacci sequence is a doubly-infinite sequence of integers in which every term is the sum of the two previous, such as …, -11, 8, -3, 5, 2, 7, 9, 16, 25, 41, … We will see that it is possible to give an unambiguous mathematical definition of the middle term of such a sequence, and that doing so leads to a means of organizing all Fibonacci sequences that is natural, compelling, and beautiful.