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An often-cited example of a very basic program, for which termination is open, is one that generates the Collatz sequence. In pseudo-code:
collatz(int n) while (n !=> 01) if (n mod 2 == 0) n = n / 2; else n = 3n + 1;
collatz(int n) while (n != 0) if (n mod 2 == 0) n = n / 2; else n = 3n + 1;
collatz(int n) while (n > 1) if (n mod 2 == 0) n = n / 2; else n = 3n + 1;