3n+1 is the first, "warm-up" problem at Programming Chalenges site:
http://www.programming-challenges.com/pg.php?page=downloadproblem&probid=110101&format=html

(This problem illustrates Collatz conjecture:
http://en.wikipedia.org/wiki/3n_%2B_1#Program_to_calculate_Collatz_sequences)

As long as the judge on this site takes only C and Java solutions, I submitted in Java some add-hock code (see at the end of this message) where I used recursion and a cache of computed cycles. Judge accepted my code and measured 0.292 sec with best overall submissions of 0.008 sec to solve the problem.

*** Question: I wonder how to implement cache for this problem in Haskell? At the moment, I am not so much interested in the speed of the code, as in nice implementation.

This is the exact problem data-memocombinators was written to solve. http://hackage.haskell.org/packages/archive/data-memocombinators/0.4.1/doc/html/Data-MemoCombinators.html

For this problem, it is too slow to memoize everything; you have to use a bounded memo table. That's why I use a combinator-based memo approach as opposed to the type-directed approach used in eg. MemoTrie. The memo table you need is something like

switch (<10^6) integral id

Luke