COLLATZ
The Collatz Sequence

COLLATZ is a MATLAB library 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 computer code and data files described and made available on this web page are distributed under the GNU LGPL license.

Related Data and Programs:

COLLATZ is available in a JAVA version and a MATHEMATICA version and a MATLAB version and a PERL version.

COLLATZ_PARALLEL is a MATLAB program which seeks the maximum Collatz sequence between 1 and N; it runs in parallel using MATLAB's "parfor" facility.

POLPAK is a FORTRAN90 library which includes the routine collatz_count for counting the length of a Collatz sequence.

SEQUENCE_STREAK_DISPLAY is a MATLAB program which can display a "streak plot" of a van der Corput sequence.

TABLE is the data format which is used for the file output by COLLATZ_WRITE.

Reference:

1. Eric Weisstein,
   "The Collatz Problem",
   CRC Concise Encyclopedia of Mathematics,
   CRC 1998.

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_write.m, writes a Collatz sequence to a file;
• timestamp.m, prints the current YMDHMS date as a timestamp;

Examples and Tests:

• collatz_test.m, runs all the tests.
• collatz_test.out, the output from the tests.
• collatz_test01.m, computes the Collatz sequence for several seeds.
• collatz_test02.m, computes the length of the Collatz sequence for seeds 1 to 100.
• collatz_test03.m, writes the Collatz sequence for seed 27 to a file, and the Collatz sequence lengths for seeds 1 to 100 to a file.

collatz_27 lists the Collatz sequence for seed 27.

• collatz_27.txt, the points.
• collatz_27.png, a PNG image of the streak plot of the sequence.

collatz_count lists the number of entries in each Collatz sequence with seeds from 1 to 100.

• collatz_count.txt, the points.
• collatz_count.png, a PNG image of the streak plot of the sequence.

You can go up one level to the MATLAB source codes.