With Simson's 1753 paper as a starting point, the current paper reports investigations of Simson's identity (also known as Cassini's) for the Fibonacci sequence as a means to explore some fundamental ideas about recursion. Simple algebraic operations allow one to reduce the standard linear Fibonacci recursion to the nonlinear Simon's recursion that is equivalent to Simson's identity and then further to a nonlinear recursion dependent only on a single preceding term. This leads to a striking nonrecursive characterization of Fibonacci numbers that is much less well-known than it should be. It is then discovered that Simson's recursion itself implies a family of linear recursions and characterizes a class of generalized Fibonacci sequences.