Abstract
The purpose of this study is to try to recognize and formerly prove the
validity of patterns associated with Collatz sequences. For each
positive integer n define h(n) to be (n/2) if n is even and 3n+1 if n
is odd. The Collatz sequence starting at n is the sequence h(n),
h^2(n), h^3(n)… It is an interesting problem to determine the behavior
of the Collatz sequence. It is a long outstanding, conjecture that for
all n, the sequence eventually becomes 4, 2, 1, 4, 2, 1… An up in the
Collatz sequence starting at n is a subsequence of the form h^k(n),
h^(k+1)(n) with h^k(n)< h^(k+1)(n). A down is defined similarly.
Through the patterns of ups and downs, we will look for patterns
concerning the number of downs in the Collatz sequence starting at n
before it goes below n. Through this research we will gain a better
understanding and appreciation for the Collatz sequence, and the
general process of observing, conjecturing, and proving results in
mathematics.