collatz

collatz, a MATLAB code which computes the Collatz sequence.

The rules for generation of the Collatz sequence are recursive. If T is the current entry of the sequence, (T is assumed to be a positive integer), then the next entry, U is determined as follows:

1. if T is 1, terminate the sequence;
2. else if T is even, U = T/2.
3. else (if T is odd and not 1), U = 3*T+1;

Although the Collatz sequence seems to be finite for every starting point, this has not been proved. Over the range of starting values that have been examined, a great irregularity has been observed in the number of entries in the corresponding sequence.

The Collatz sequence is also known as the "hailstone" sequence or the "3n+1" sequence.

Licensing:

The information on this web page is distributed under the MIT license.

Languages:

collatz is available in a MATLAB version and an Octave version and a Python version.

Related Data and Programs:

collatz_test

collatz_polynomial, a MATLAB code which implements the Collatz polynomial iteration, a polynomial analog of the numerical iteration that is also known as the 3n+1 conjecture or the hailstone sequence.

collatz_recursive, a MATLAB code which demonstrates recursive programming by considering the simple Collatz 3n+1 problem.

polpak, a MATLAB code which evaluates a variety of mathematical functions, polynomials, and sequences, including Bell, Benford, Bernoulli, Bernstein, Cardan, Catalan, Charlier, Chebyshev, Collatz, Delannoy, Euler, Fibonacci, Gegenbauer, Gudermannian, Hermite, Hofstadter, Jacobi, Krawtchouk, Laguerre, Lambert, Legendre, Lerch, Meixner, Mertens, Moebius, Motzkin, Phi, Sigma, Stirling, Tau, Tribonacci, Zernike.

Reference:

1. Martin Gardner,
   "Slither, 3X+1 and Other Curious Questions",
   Wheels, Life, and Other Mathematical Amusements,
   WH Freeman, 1983.
2. Brian Hayes,
   "On the ups and downs of hailstone numbers",
   Scientific American,
   January 1984,
   Volume 250, Number 1, pages 10-16.
3. Patrick Honner,
   "The Simple Math Problem We Still Can't Solve",
   Quanta,
   22 September 2022.
4. Eric Weisstein,
   "The Collatz Problem",
   CRC Concise Encyclopedia of Mathematics,
   CRC Press, 2002,
   Second edition,
   ISBN: 1584883472,
   LC: QA5.W45.

Source Code:

• collatz.m, computes the Collatz sequence for a given starting point;
• collatz_count.m, returns the number of entries in the Collatz sequence for a given starting point;
• collatz_inverse.m, returns the preimage or "inverse" of a set of values under the Collatz transformation;
• collatz_level.m, returns the values that are k steps away from 1.
• collatz_max.m, returns the maximum entry in the Collatz sequence for a given starting point;
• collatz_write.m, writes a Collatz sequence to a file;
• lollatz_permutation.m, applies the Lollatz permutation to a given set of values.
• lollatz_sequence.m, computes a Lollatz sequence that stops if it repeats, or if it exceeds a maximum number of steps.
• mollatz.m, computes the Mollatz 3N-1 sequence for a given starting point;
• nollatz.m, computes the Nollatz N+1 sequence for a given starting point;