Chapter 8.2, Problem 103E

The hailstone sequence Here is a fascinating (unsolved) problem known as the hailstone problem (or the Ulam Conjecture or the Collatz Conjecture). It involves sequences in two different ways. First, choose a positive integer N and call it a₀. This is the seed of a sequence. The rest of the sequence is generated as follows: For n = 0, 1, 2, …

$a_{n+1}=\{\begin{array}{cc}{a}_n/2& \text{if}{a}_n\text{is}\text{even}\\ 3a_n+1& \text{if}{a}_n\text{is}\text{odd}.\end{array}$

However, if a_n = 1 for any n, then the sequence terminates.

1. a. Compute the sequence that results from the seeds N = 2, 3, 4, …, 10. You should verify that in all these cases, the sequence eventually terminates. The hailstone conjecture (still unproved) states that for all positive integers N, the sequence terminates after a finite number of terms.
2. b. Now define the hailstone sequence {H_k}, which is the number of terms needed for the sequence {a_n} to terminate starting with a seed of k. Verify that H₂ = 1, H₃ = 7, and H₄ = 2.
3. c. Plot as many terms of the hailstone sequence as is feasible. How did the sequence get its name? Does the conjecture appear to be true?