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Max-Minus-Min Sequence Analysis

This tool is for generating, visualizing, and analyzing max-minus-min sequences (MMM sequences) based on various initial conditions. An MMM sequence is always defined by a(0) = x, a(1) = y, a(2) = z, and a(n) = max(a(n-1), a(n-2), a(n-3)) - min(a(n-1), a(n-2), a(n-3)) = max(previous 3 terms) - min(previous 3 terms). For example, one MMM sequence might go [1, 2, 3, 2, 1, 2, 1, 1, 1, 0, 1, 1, 1 ...].

I have been exploring the properties of these sequences as a fun side project. For instance, one nice property is that if S is an MMM sequence, then so is a*S for any a. Using our example above, then we can be sure that [2,4,6,4,2 ...] is a valid MMM sequence using a=2.

Setup

  1. Clone the repository:
git clone https://github.com/jackaldenryan/mmm-sequence.git
cd mmm-sequence
  1. Install Python 3.11 and tkinter (required for plotting):
# On MacOS
brew install [email protected]

# On Ubuntu/Debian
sudo apt-get install python3.11 python3.11-tk

# On Windows
# Download Python 3.11 from python.org (tkinter is included)
  1. Install Graphviz (required for tree visualization):
# On MacOS
brew install graphviz

# On Ubuntu/Debian
sudo apt-get install graphviz

# On Windows
# Download from https://graphviz.org/download/
  1. Create and activate a virtual environment:
python3.11 -m venv venv
source venv/bin/activate  # On Unix/MacOS
# OR
venv\Scripts\activate  # On Windows
  1. Install dependencies:
pip install -r requirements.txt

Usage

The functionality is provided through a Jupyter notebook interface. To start, run:

jupyter notebook mmm_sequence/main.ipynb

The notebook contains the following cells, each corresponding to a different analysis function:

Plot Single Sequence

Generate and plot a max-minus-min sequence with specified initialization. Parameters:

  • init: List of 3 integers for initial values
  • max_points: (Optional) Maximum number of points to generate. If not specified, plots until convergence.

Example:

init = [5, 7, 11]
max_points = 20  # optional
plot_seq(init, max_points)

Plot Multiple Sequences

Generate and plot multiple MMM sequences using different initialization methods:

  1. Random initialization:

    • n: Number of sequences to generate
    • max_val: Maximum value for random initialization

  2. As_Ds initialization (init = [a, a+d, a+2*d]):

    • min_a/max_a: Range for parameter a
    • min_d/max_d: Range for parameter d

  3. Xs_Ys_Zs initialization (init = [x, y, z]):

    • min_x/max_x: Range for x values
    • min_y/max_y: Range for y values
    • min_z/max_z: Range for z values

3D Convergence Visualization

Create interactive 3D plots showing how sequences converge. Parameters:

  • type: Either 'time' (steps until convergence) or 'value' (final value before convergence)
  • min_a/max_a: Range for parameter a
  • min_d/max_d: Range for parameter d

The resulting plot shows fascinating fractal-like patterns in the convergence behavior.

2D Convergence Analysis

Create 2D plots showing convergence behavior while varying one parameter. Parameters:

  • type: Either 'time' or 'value'
  • vary_param: Choose 'a' or 'd' to vary
  • min_vary/max_vary: Range for the varying parameter
  • fixed_val: Value for the fixed parameter

Seed Sequence Analysis

Analyze MMM sequences derived from sliding windows over seed sequences. Available seed functions:

  • primes
  • naturals
  • randoms
  • randoms_inc
  • odds
  • odds_random
  • odds_skip
  • primes_random
  • primes_even
  • primes_odd

Parameters:

  • seq_func: Name of the seed function to use
  • n: Length of seed sequence

Backwards Tree Generation

Visualize all possible paths that could lead to a given sequence end. Parameters:

  • end: List of 3 integers representing the end of an MMM sequence
  • n: Number of backward steps to generate
  • negatives: Whether to include paths through negative numbers

The visualization highlights nodes in green if their value is not a multiple of the sequence's convergence value. Three consecutive white nodes indicate all subsequent nodes must also be white.

Example:

end = [20, 10, 10]
n = 5
negatives = False
gen_backwards_tree(end, n, negatives)

Each visualization is automatically saved in the backwards_trees directory with a timestamp.