### Abstract:

A study of the running time of several known algorithms and several new algorithms to compute the n[superscript th] element of the Fibonacci sequence is presented. Since the size of the n[superscript th] Fibonacci number grows exponentially with n, the number of bit operations, instead of the number of integer operations, was used as the unit of time. The number of bit operations used to compute f[subscript n] is reduced to less than 1/2 of the number of bit operations used to multiply two n bit numbers. The algorithms were programmed in Ibuki Common Lisp and timing runs were made on a Sequent Balance 21000. Multiplication was implemented using the standard n² algorithm. Times for the various algorithms are reported as various constants times n². An algorithm based on generating factors of Fibonacci numbers had the smallest constant. The Fibonacci sequence, arranged in various ways, is searched for redundant information that could be eliminated to reduce the number of operations. Cycles in the b[superscript th] bit of f[subscript n] were discovered but are not yet completely understood.