Consider the function T (n) defined on the positive integers as follows. If n is even, T (n) = n/2. If n is odd, T (n) = 3n + 1. The Collatz Conjecture states that for any integer n, the sequence n, T (n), T (T (n)), . . . will eventually reach 1. We consider several generalizations of this function, focusing on functions which replace "3n + 1" with "3n + b" for odd b. We show that for all odd b < 400, and all integers n ≤ 106, iterating this function always results in a finite cycle of values. Furthermore, we empirically observe several interesting patterns in the lengths of these cycles for several classes of values of b.Faculty Sponsor: Dr. Dominic Klyve

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