xThere's prior art to this in the work of Gauss, who remarked on the complexity of the iteration,[1]_{n+1}= 4x_{n}(1 - x_{n})

xA random number generator that was popular in the early computing days of slow clock rates and small memories is an iteration called the linear congruential generator (LCG),_{n+1}= FractionalPart[1/x_{n}]

xCommon values for the LCG parameters are m = 2_{n+1}= (ax_{n}+ b) mod m

xThe logistic map produces a chaotic sequence for r values between 3.57 and 3.83._{n+1}= rx_{n}(1 - x_{n})

1) Start with a natural number, nAs you can see, step 2 will always decrease n, and step 3 will always increase n, so you can expect some random up-down action. What you don't expect is that you always terminate in n = 1, no matter the value of n_{0}

2) If n is even, then n_{i+1}= n_{i}/2

3) If n is odd, then n_{i+1}= 3n_{i}+ 1

4) Continue at step (2) until n = 1

27 → 41 → 62 → 31 → 47 → 71 → 107 → 161 → 242 → 121 → 182 → 91 → 137 → 206 → 103 → 155 → 233 → 350 → 175 → 263 → 395 → 593 → 890 → 445 → 668 → 334 → 167 → 251 → 377 → 566 → 283 → 425 → 638 → 319 → 479 → 719 → 1079 → 1619 → 2429 → 3644 → 1822 → 911 → 1367 → 2051 → 3077 → 4616 → 2308 → 1154 → 577 → 866 → 433 → 650 → 325 → 488 → 244 → 122 → 61 → 92 → 46 → 23 → 35 → 53 → 80 → 40 → 20 → 10 → 5 → 8 → 4 → 2 → 1

908 → 454 → 227 → 341 → 512 → 256 → 128 → 64 → 32 → 16 → 8 → 4 → 2 → 1All this adds to the mystery of this sequence. This interesting recursion was discovered in 1932 by Lothar Collatz, who was at the time a twenty year old mathematics student. As discussed in an entertaining book of number anecdotes by George Szipiro,[2] the Collatz conjecture has had many names over the years before reverting to that of its discoverer. It was popularized in the US by Stanislaw Ulam, who introduced it to the extremely talented corp of people who worked with him on the Manhattan Project. For a time after that it was known as Ulam's problem. Then it became known as the Hailstone sequence because of the similarity of the up-down motion of its numbers to the process of hailstone formation. It was also known as the Kakutani problem, after another mathematician who worked on the conjecture. We shouldn't be too surprised at this, since it happened in the days before computer-assisted search.

- Stephen Wolfram, "A New Kind of Science," Wolfram Media, May 14, 2002, page 918.
- George G. Szipiro, "The Secret Life of Numbers," Joseph Henry Press (Washington, D.C., 2006), Chapter 6 ("A Puzzle by Any Other Name"). pp. 20-23 (Via Amazon).